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index 000000000..c74f31816
--- /dev/null
+++ b/01.curriculum/01.physics-chemistry-biology/01.niv1/frontmatter.yaml
@@ -0,0 +1,9 @@
+# This page configuration is shared by all locales, in this directory.
+# You can override it in the individual frontmatter of the pages.
+
+content:
+    # https://learn.getgrav.org/15/content/collections#summary-of-collection-options
+    items: @self.children
+
+anchors:
+    active: false
\ No newline at end of file
diff --git a/01.curriculum/01.physics-chemistry-biology/01.niv1/topic.es.md b/01.curriculum/01.physics-chemistry-biology/01.niv1/topic.es.md
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index 000000000..99057209a
--- /dev/null
+++ b/01.curriculum/01.physics-chemistry-biology/01.niv1/topic.es.md
@@ -0,0 +1,5 @@
+---
+title: llanuras
+media_order: sesituersynt_400_2400.jpg
+---
+
diff --git a/01.curriculum/01.physics-chemistry-biology/01.niv1/topic.fr.md b/01.curriculum/01.physics-chemistry-biology/01.niv1/topic.fr.md
new file mode 100644
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--- /dev/null
+++ b/01.curriculum/01.physics-chemistry-biology/01.niv1/topic.fr.md
@@ -0,0 +1,4 @@
+---
+title: plaines
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--- a/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/01.N2-geometrical-optics-laws/textbook.en.md
+++ /dev/null
@@ -1,15 +0,0 @@
----
-title: 'The 4 laws of the geometrical optics'
----
-
-Law 1
-: In an **homogeneous medium**, the *light rays travel in straight lines*
-: Dans un **milieu homogène**, la *lumière se propage en ligne droite*
-
-!!! EXAMPLE    of homogeneous medium : vacuum, air, glass, water,  liquids, ...
-!!! EXEMPLE    de milieux homogènes : le vide, l'air, le verre, l'eau, les liquide, ...
-
-Law 2
-:Dans le vide, l'énergie de la lumière se propage en suivant les rayons lumineux à la vitesse $`c=300000 km/s = 3.10^8 m/s`$.
-: In vaccum, the energy of the light travel along the light rays at the speed $`c=300000 km/s = 3.10^8 m/s`$.
-!!! COMMENT the value of the speed in vaccuum, 3.10^8 m/s is the approximated value you have to remember. The speed of the light is not anymore measured, but fixed at the exact value of 
diff --git a/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/01.N2-geometrical-optics-laws/topic.fr.md b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/01.N2-geometrical-optics-laws/topic.fr.md
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--- a/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/01.N2-geometrical-optics-laws/topic.fr.md
+++ /dev/null
@@ -1,15 +0,0 @@
----
-title: 'Les 4 lois de l''optique géométrique'
----
-
-Law 1
-: In an **homogeneous medium**, the *light rays travel in straight lines*
-: Dans un **milieu homogène**, la *lumière se propage en ligne droite*
-
-!!! EXAMPLE    of homogeneous medium : vacuum, air, glass, water,  liquids, ...
-!!! EXEMPLE    de milieux homogènes : le vide, l'air, le verre, l'eau, les liquide, ...
-
-Law 2
-:Dans le vide, l'énergie de la lumière se propage en suivant les rayons lumineux à la vitesse $`c=300000 km/s = 3.10^8 m/s`$.
-: In vaccum, the energy of the light travel along the light rays at the speed $`c=300000 km/s = 3.10^8 m/s`$.
-!!! COMMENT the value of the speed in vaccuum, 3.10^8 m/s is the approximated value you have to remember. The speed of the light is not anymore measured, but fixed at the exact value of 
diff --git a/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/01.N2-geometrical-optics-laws/topics.es.md b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/01.N2-geometrical-optics-laws/topics.es.md
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--- a/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/01.N2-geometrical-optics-laws/topics.es.md
+++ /dev/null
@@ -1,4 +0,0 @@
----
-title: 'a renommer, pas ca, ici, couplage ondulatoire photons, rays '
----
-
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@@ -1,5 +1,5 @@
 ---
-title: 'new course : main'
+title: 'The 5 laws of light rays optics : main'
 ---
 
 new course : main
\ No newline at end of file
diff --git a/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/02.optical-phenomena/02.new-course-overview/cheatsheet.en.md b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/02.optical-phenomena/02.new-course-overview/cheatsheet.en.md
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--- a/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/02.optical-phenomena/02.new-course-overview/cheatsheet.en.md
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@@ -1,5 +1,76 @@
 ---
-title: 'new course  : overview'
+title: 'Wave or corpuscules? or both ...'
+media_order: 'corpuscule-trajectory-tennis-ball.jpg,refraction-cuillere.jpg,sky-light-rays-827.jpg,wave-corpuscular-nature-of-light-experiment1.gif,waves-interferences-surface-water-827-496.jpg,reflexion-statuette.jpg'
 ---
 
-new course  : overview
\ No newline at end of file
+### How do we understand our physical world ?
+
+* We interpret the world with two intuitive concepts that are in our mind mutually exclusive : the concept of corpuscule and the concept of wave
+
+##### Localized corpuscules with trajectories
+
+In our mind, a **corpuscule** is *localized in space*, and **At each instant** can be characterized by :
+* a *position in space*
+* a *speed*
+* an *acceleration*
+
+It carries :
+* *matter*
+* *energy*
+* *momentum*
+
+Between two instants, its movement describes a *trajectory* : continuous set of positions followed by the corpuscule between these two instants.
+
+The **behavior of a corpuscule** *at human scale*, and **their interactions with each other**, is well described and predicted by *classical mechanics*.
+
+* **classical mechanics** = **Newton's mechanics** = **point mechanics**
+
+![](corpuscule-trajectory-tennis-ball.jpg)
+
+##### Waves that interfer to each other
+
+In our mind, a **wave**  is *not localized in space*, and **in the whole time** can be characterized by :
+* a *frequency*
+* a *velocity*
+* frequency + velocity $\Longrightarrow$ a *wavelength*
+* an *amplitude*
+* a *polarization* (depending of wave type)
+
+A wave 
+* *does not carry matter*
+* *carries energy*
+
+**Several waves** interfer with each other, giving rise to *interferences*
+
+A wave **passing through a small aperture** (of size of the order or small than wavelength) is *diffracted*
+
+![](waves-interferences-surface-water-827-496.jpg)
+
+
+### How light behaves ?
+
+![](wave-corpuscular-nature-of-light-experiment1.gif)
+
+A modern Youngs'type experiment :
+behavior of light passing through two slits and 
+The light emerging from a double slit, whose the width (of each slot) and the spacing (between the two slots) are sufficiently small, 
+
+
+### What optical phenomena do we observe in everyday life ?
+
+* The **light** describes in *straight lines in air*.<br>
+$\Longrightarrow$ concept of *light ray*
+
+![](sky-light-rays-827.jpg)
+
+##### Refraction phenomenon
+
+* The light seems to *change direction from water to air*.<br>
+![](refraction-cuillere.jpg)
+
+* can be d
+
+##### Reflection phenomenon
+
+![](reflexion-statuette.jpg)
+
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diff --git a/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/03.optical-systems-vergence/02.optical-systems-vergence-overview/cheatsheet.en.md b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/03.optical-systems-vergence/02.optical-systems-vergence-overview/cheatsheet.en.md
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--- a/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/03.optical-systems-vergence/02.optical-systems-vergence-overview/cheatsheet.en.md
+++ b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/03.optical-systems-vergence/02.optical-systems-vergence-overview/cheatsheet.en.md
@@ -1,5 +1,404 @@
 ---
 title: 'new course  : overview'
+media_order: 'objeto-fisico-imagen-virtual-haz-3-rayos-bb.jpg,objeto-fisico-imagen-virtual-pantalla-bb.jpg,objeto-imagen-posicion-tamano-bb.jpg,objeto-imagen-bb.jpg,objeto-imagen-3-rayos-bb.jpg,plano-imagen-real-convergencia-bb.jpg,plano-imagen-real-convergencia-pantalla-bb.jpg,plano-objeto-fisico-bb.jpg,vision-object-b.jpg,imagerie-rays-optics.gif,vision-object-image-2.jpg,vision-image.jpg,axe_opt.gif,plano-objeto-plan-image-bb.jpg,objeto-imagen-haz-3-rayos-bb.jpg,objeto-fisico-imagen-virtual-haz-3-rayons-pantalla-bb.jpg,plano-objeto-plano-imagen-real-pantalla-bb.jpg,sym_rev_2.gif,imagerie-rayons-lens-1.jpg,imagerie-rayons.jpg,imagerie-rays-optics.jpg'
 ---
 
-new course  : overview
\ No newline at end of file
+<!--To be organized diferently, in is not study of "reality" and model it with physical principles and laws, it is more linked with technology : what is an image and how to do it : so remains unpublished
+
+### What is an image ?
+
+An image is a way to restore for the view part of the shape and color of objects or beings, of a scene of everyday life or a landscape. .... Oulà, it is a very good question indeed ! ok, some ideas :
+
+#### Real 3D imaging 
+
+In real 3D imaging, the image itself is in real 3D, part of the information of the three dimensional shape of the object is kept : if we move a little, the viewing angle of the body or the scene is changing. So, two people watching the same "real 3D image" see the body from different angles, they don't have the same scene in their field of view.
+
+$\Longrightarrow$ no  conflicting information received by our natural biological means of determining distances: eye accommodation and binocular vision.
+
+Technics for real 3 dimensional imaging :
+
+* optics imaging : based on refraction and reflexion phenonema induced by macroscopical optical systems. The 3D image requires the presence of the object, it can not be stored. It is widely used to store specific 2D fields of view of the bodies or scenes. For this, the energy distribution of the light, in a limited number of spectral range (the 3 primary colors of the additive synthesis in the visible)) in a 2D plane (the plane of the imaging sensor)...blabla bla
+* holography : based on diffraction phenomenon, the 3D information of the bodies can be stored in the 2D plane of an holographic film. High resolution papers... blablabla lambda / 10...
+* 3D technics : required eye glasses : an diffent image is provided at each eye, calculated in real time by a powerfull computer, from the information about the 3D shape of bodies in the field of view and the real time information of the relative position of the spectator or gamer relative to the various bodies in the scene.
+
+Technics for false 3 dimensional imaging :
+* 3D cinema technics : all viewers see the same 3D image.
+
+#### False 3D imaging : 3D-cinema 
+
+#### 2D imaging : photography and 2D-cinema 
+
+A whole chapter to write $\Longrightarrow$ nature of light (refraction, reflexion, diffraction and interferences), geometry, Radon, computing, electronics ...
+-->
+
+### What is optical imaging ?
+
+#### Objective
+
+* **Optical** $\Longrightarrow$ *visible* range + *near infrared* + *near UV*
+
+* to **realize optical images of physical objects** by the use of *simple optical elements* which can be combined in *optical systems* to form *optical instruments*.
+
+#### Physical object
+
+* **Physical object** : *large* (compared to $\lambda_{optical}$) *volume* of matter (liquid or solid) whose *external surface breaks down in a huge number of microscopic surfaces*.
+
+* **Physical object point** = **point source** : <br>
+\- *microscopic surface* part of the overall surface of the physical object.<br>
+\- *emits* or *diffuses light in all direction* outside the volume. That means in equivalent ways :  emits *spherical waves* (_wave optics_), emits **light rays** (_rays optics_), emits *photons* (_photons optics_) that **diverge from the object point**.
+
+* Point source **pencil of light** = **bundle of rays** : part of the *__light emitted by a point source__* that *intercept an optical system* or *pass through a limiting aperture*.
+<!-- In the main part, make the difference between pencil or bundle of rays, and the beam of an extended source. In optics, only pencils are really importants.
+And to say in a warning taht if the difference is made in term between pencil and beam in English, it is not made in French (faiscaeu). In Spanish?-->
+
+* **naked eyes** = **direct vision** of an object : *pencils* from all visible point sources of the object *intercept the pupil of my eyes*.
+
+![](vision-object-b.jpg)<br>
+_Direct vision : the pencils of each visible point souces intercept the iris of my eyes_
+
+#### Optical image
+
+* Object seen from a **specific angle of view**, whose principal direction named **line of sight** is (_when oriented positively in direction of the eye_) the **optical axis** of the imaging system.
+
+* Part of the *light that diverge from any point source* of the seen object, has to *converge back in a new location* in space named **image point**. 
+
+* **Image** : *set of all the image points*. 
+ 
+* **form consistency** between initial object and its image, but *shape distortions may appear*.
+
+![](vision-image.jpg)<br>
+_Image vision : an optical imager (rectangle) has modified the incident pencils. Only pencils from image points enter my eyes. I don't see anymore the initial extended object._
+
+
+#### Optical imager and basis physical principles.
+
+** Imager** :
+* *intercept part of the light* emitted or diffused by the physical object.
+* *modify the pencils* of light from each object point *to converge them back* into a new location in space.
+
+** Optical imager** :
+* create a **real three dimensional** *image* of the extended object surface oriented towards the imager.
+* use **refraction and/or reflexion** phenomena.
+<!-- In main or beyond, talk about this, in a comment. Term "optical", contradiction. "visible?" or "refraction/reflexion use?". Holographic image work in the visible range too, but is based on diffraction pehnomenon. 
+for science : optics refer specifically to the visible range.
+for technology : same... but in geométrical optics, we consider only imager bse on refraction and reflexion phenomena-->
+* Imagers can be : individual **thin simple optical elements** or **centered optical systems**
+
+![](vision-object-image-2.jpg)<br>
+_By use of refraction and/or reflexion phenomena, an optical imager modifies all incident pencils to converge towards image points._
+
+#### Thin simple optical element
+
+*  often has a **symetry of revolution about an axis**.<br>
+
+![](sym_rev_2.gif)<br>
+_simple optical element : refracting or reflecting element, rotationaly symmetrical around an axis_
+
+* **Thin** $\Longrightarrow$ *diameter $\gg$ thickness or depth*)
+
+* **Simple** : surfaces of *simple* optical element are *plane or spherical*
+
+* **Thin optical elements** studied are :<br>
+\- *plane or thin curved refracting surfaces*.<br>
+\- *plane or thin curved mirrors*.<br>
+\- *thin lenses*.
+
+#### Centered optical systems
+
+* **Combination** *of thin simple optical elements* **centered** *on a common axis*  that becomes the optical axis of the system (_when positively oriented in direction of the incident light on the system)._
+
+* *Interest* : can be **characterized as a whole**.
+
+![](axe_opt.gif)<br>
+_optical system : combination of thin simple optical elements, centered on a same optical axis_
+
+### What physical framework to describe optical imaging ?
+
+#### From idealization to physical and usefull reality
+
+* **Point** : mathematical concept of vanishing volume.<br>
+has a *location* in space, but *no extension*, *no orientation*.
+
+* **Image point** physical meaning :  the pencil emerging from the imager focuses on a so small volume that its *extension can be neglected*.<br>
+\- extension of the volume can not be resolved naked eye vision.<br>
+\- surface illumated in the sensor plane is below the size of a pixel.
+
+* perfectly **stigmatic optical system** : *gives one image point for each object point* (**don't exist**).
+
+* **quasi stigmatic optical system** : *under certain conditions of use* a set of optical elements is *quasi-stigmatic* and so becomes an *imager*.
+
+* **Optical imager** = **quasi-stigmatic optical element or system** used to give images.
+
+<!-- image required, to explain difference between a quasi-stagmatic and a no stigmatic optical element-->
+
+#### Framework of light rays optics.
+
+* We use the concept of **light rays**, coming from *light rays optics*
+
+* **light rays optics** = **geometrical optics**
+
+* A *light pencil that diverges from a point source*, is modified by an imager and *converges back towards a image point*.<br><br>
+&nbsp;&nbsp;equivalent to <br><br>
+\- *All rays emerging from a point source* are deviated by the imager and *cross back on the conjugated image point*.<br><br>
+$\Longrightarrow$  **knowledge of only two different rays** from a point source through the imager **is  sufficient** to determine image position.
+
+* For any object point of any imager, **trajectories of 3 specific rays will be specified**.
+
+![](imagerie-rays-optics.gif)<br>
+_Light rays optics : 3 specific ray are specified (2 are sufficient) to locate the image point of any object point (in this figure, the thin imager is a thin lens)_
+
+
+### How is modeled and characterized a thin simple optical element in light rays optics?
+
+#### Thin simple optical element
+
+* **Thin simple optical element** = optical element  whose *thickness can be neglected in front of diameter* **$\Longrightarrow$ represented by a plane (Elt)**.
+
+* Thin simple optical elements* are **rotationally symetrical around an axis** :<br>
+$\Longrightarrow$ same optical behavior in all plane containing the axis of symetry.<br>
+$\Longrightarrow$ *object point and conjugate image point belong to a same plane that contains the axis of symetry.<br>
+$\Longrightarrow$ working in the **sectional view** corresponding to that plane is **sufficient**.
+
+* **Representation** on a sectional view containing the optical axis: *straight line [Elt] perpendicular to the optical axis*.
+
+* *Location* with its **vertex S** : intersection of [Elt] with the optical axis.
+
+#### Optical behavior
+
+* \- At each object point B corresponds to a unic image point B'.<br>
+\- At each image point B 'corresponds a unic object point B :<br>
+$\Longrightarrow$ **B and B' are conjugate points**.
+![](objeto-imagen-bb.jpg)
+
+* All object point (A ; B, C, ...) of an object plane (PO) perpendicular to the optical axis have conjugated points images (A' ; B', C', ...) in a same image plane (PI) perpendicular to the optical axis :<br>
+$\Longrightarrow$ **(PO) and (PI) are conjugate planes**.
+![](plano-objeto-plan-image-bb.jpg)
+
+#### Coordinates to locate object and image points
+
+* Each point (object and image) is *projected perpendicularly on the optical axis* :<br>
+\- point source B $\Longrightarrow$ point A on the optical axis.<br>
+\- conjugate image point B' $\Longrightarrow$ point A' on the optical axis.<br>
+
+* *Distance of a point from thin imager* : **algebraic distance between imager vertex and point projection**.<br>
+ \- distance of point source B from imager [Elt] :  **$\overline{SA}$**<br>
+\- distance of conjugate image point B' from imager [Elt]  :  **$\overline{SA'}$**
+
+* *Distance of a point from optical axis* : **algebraic distance between point projection and point itself**.<br>
+ \- distance of point source B from axis [Elt] :  **$\overline{AB}$**<br>
+\- distance of conjugate image point B' from axis :  **$\overline{A'B'}$**
+
+![](objeto-imagen-posicion-tamano-bb.jpg)
+
+#### Characterization of a thin simple optical element 
+
+**4 points ** * located on the optical axis* that characterize optical behavoir : **S , C, F and F**
+
+* **S** : **vertex** of the thin imager : *indicates its position* in space, and on the optical axis.
+*  **C** : **nodal point** : by definition *all rays* (or its extension) that *pass through nodal point C* has *unchanged direction* when leaving the thin optical element. Position characterizes by its algebraic distance from vertex S : $\overline{SC}$.<br>
+The *nodel point* is a *center* (_whose exact physical meaning depends of the type of thin simple optical element_)
+
+*   **F'** : **image focal point** = **second focal point** = **image focus** : *incident rays* (or their extensions) *parallel to the optical axis* (or their extensions), *after leaving* the thin imager, *pass through F'*.
+*   **F** : **object focal point** = **first focal point** = **object focus** : *incident rays* (or their extensions) *passing through F leave* (or their extensions) the imager *parallel to the optical axis*.
+
+which define **3 important planes, perpendicular to the optical axis**
+
+* **(Elt)** : representation of the **thin imager interface, contains S** : *incident rays* (or their extension) *change of direction when passing through (Elt)*.
+* **(P')** : **image focal plane** = **second focal plane , contains F'** : *all incident rays parallel to each other* originated to a unic point at infinity *leave (Elt) to converge* (or their extension) *on a unic image point B' located in (P')*. Location of B' in (P') is the *intersection of the ray passing through C with (P')*.
+* **(P)** : **object focal plane** = **first focal plane , contains F** : *all incident rays originated from a unic point source B leave (Elt) parallel to each other*, given an image point B' located at infinity, *in direction of the ray passing through C*.
+
+and **2 important algebraic distances**
+
+* **$f'=\overline{SF'}$** : *algebraic distance from thin imager (Elt) to image focal plane (P)* :<br>
+*$f'=\overline{SF'}$* **$>0 \Longleftrightarrow$ converging** thin imager.<br>
+*$f'=\overline{SF'}$* **$<0 \Longleftrightarrow$ diverging** thin imager.
+
+* **$f=\overline{SF}$** : *algebraic distance from thin imager (Elt) to object focal plane (P)*.
+
+
+### Important significations of focal planes
+
+#### Image focal plane :
+
+* *Physical sense* :<br> **object B at infinity $\Longleftrightarrow |\overline{SB}| \ll |f|$**.
+
+* **Object at infinity (P) $\Longleftrightarrow$ object in (P')**, can be **viewed by naked eye** whether *convergent or divergent* optical element.
+
+![](plano-imagen-real-convergencia-bb.jpg)<br>
+_example : direct vision of the universe through a telescope (telescope is not a thin imager, but same image focal plane defintion)_
+
+* **Object at infinity (P) $\Longleftrightarrow$ object in (P')**, to be **captured by an image sensor** *if convergent optical element*.
+* 
+![](plano-imagen-real-convergencia-pantalla-bb.jpg)<br>
+_example : picture taken with a telephoto lens (telephoto lens is not a thin imager, but same image focal plane defintion)_
+
+#### Object focal plane (P)
+
+* *Physical sense* :<br> **image B' at infinity $\Longleftrightarrow |\overline{SB'}| \ll |f'|$**.
+
+* **Object in (P) $\Longleftrightarrow$ image at infinity**
+
+![](plano-objeto-fisico-bb.jpg)<br>
+(_examples : Object can be the lightbulb of a lighthouse or a headlight, the film in a film projector_)
+
+### How to determine the image given by a thin simple optical element ?
+
+#### Graphical study
+
+##### Two different scales <!--of image point-->
+
+* Object and image :  *tranverse sizes $|\overline{AB}|$, $|\overline{A'B'}\;\;\ll$ distances from optical element* along optical axis $|\overline{SA}|$ , $|\overline{SA'}|$.<br>
+$\Longrightarrow$ dimensions perpendicular to optical axis $\ll$ dimensions along optical axis 
+
+* So **Accurate graphical study** $\Longrightarrow$ *greatly magnify scale perpendicular to optical axis*.
+
+![](objeto-imagen-posicion-tamano-bb.jpg)
+
+##### Determining of conjugate points <!--of image point-->
+
+From given point (object or image) **3 specific light rays can be drawn** ( only 2 required), whose *intersection gives the conjugate point* :
+
+![](objeto-imagen-3-rayos-bb.jpg)
+
+* **[Ray1]** : Incident rays (or their extensions) passing through the object focal point F leave (or their extensions) the imager parallel to the optical axis.
+* **[Ray2]** : Incident rays (or their extensions) parallel to the optical axis pass (or their extensions), after leaving the thin imager, through the image focal point F'.
+* **[Ray3]** : Incident ray (or its extension) passing through point C has unchanged direction when leaving the thin imager.
+
+##### Consequences from focal points definitions
+
+* **Incident rays parallel to each other** intersect on a *same point in the image focal plane* (P').<br>
+(location : intersection of (P') with [Ray3])
+
+* **Emergent rays parallel to each other** diverge from a *same point source located in the object focal plane* (P).<br>
+(location : intersection of (P) with [Ray3])
+
+
+##### Determining the emergent ray corresponding to any incident ray
+
+* **deflection of a ray by a thin optical element** : *local interaction the point of impact* (refraction or reflection) between  incident ray and thin optical element $\Longrightarrow$ *independent of the distance of the point source*.
+ 
+* so consider **point source B to infinity** $\Longrightarrow$  image point *B' would be in the image focal plane (P')*.
+
+* **ray from B** (to infinity) **passing through point C** has *unchanged direction* $\Longrightarrow$ *location of B' in (P')*.
+
+* **Emerging ray** (or its extension) = *straight line between impact point and B'*.
+
+##### Determining the incident ray corresponding to any emergent ray
+
+* **deflection of a ray by a thin optical element** : *local interaction the point of impact* (refraction or reflection) between  emergent ray and thin optical element $\Longrightarrow$ *independent of the distance of the image point*.
+ 
+* so consider **image point B' to infinity** $\Longrightarrow$  point source *B would be in the object focal plane (P)*.
+
+* **emerging ray towards B** (to infinity) **passing through point C** has *unchanged direction* $\Longrightarrow$ *location of B in (P)*.
+
+* **incident ray** (or its extension) = *straight line between impact point and B*.
+
+
+#### Analytical determining of conjugate points <!--of image point-->
+
+##### Distance of the conjugate point from thin optical element
+
+* deduced from the **thin optical element equation**
+
+* **thin optical element equation** = **thin optical element formula**
+
+* will be given for :<br>
+\- *thin spherical mirror equation*<br>
+\- *thin spherical refracting surface equation*<br>
+\- *thin lens equation*<br>
+(_will be demonstrated in level foothills_)
+
+##### Distance of the conjugate point from optical axis
+
+* deduced from the **transverse magnification $M_T$**.
+
+* **transverse magnification** = **lateral magnification**
+
+* **Definition** : **$M_T=\dfrac{\overline{A'B'}}{\overline{A'B'}}$**
+
+* **$M_T>0$** *$\Longleftrightarrow$* **erect image**<br>
+  **$M_T<0$** *$\Longleftrightarrow$* **inverted image**
+
+* **Expression** : *depends on* type of *simple optical element* :<br>
+ will be given for :<br>
+\- *thin spherical mirror equation*<br>
+\- *thin spherical refracting surface equation*<br>
+\- *thin lens equation*
+
+* **algebraic value of $M_T$** : is a *function of $\overline{AB}$ and $\overline{A'B'}$*<br>
+$\Longrightarrow$ depends on conjugate points locations<br>
+$\Longrightarrow$ **does not characterized the optical element** itself.
+
+### How to characterize the action of an imager ?
+
+##### Characterization of an extended object
+
+* Extended object [AB] or [B$_1$B$_2$], **perpendicularly to the optical axis** :<br><br>
+\- characterized by the *algebraic transverse size* (distance between its extremities) : **$\overline{AB}$** or **$\overline{B_1B_2}$**<br><br>
+\- or characterized by the *apparent angle* (in algebraic value or not) *substended by the object at nodal point* of the eye (direct vision) or nodal point of the thin imager : **$\alpha$** or **$\overline{\alpha}$**
+
+* Extended object [A$_1$A$_2$] **along the optical axis** :<br><br>
+ characterized by the *algebraic longitudinal size* (distance between its extremities) : **$\overline{A_1A_2}$** 
+
+##### Characterization of its conjugate extended image
+
+* Extended image [A'B'] or [B'$_1$B'$_2$], **perpendicularly to the optical axis** :<br><br>
+\- characterized by its *algebraic size* (distance between its extremities) : **$\overline{A'B'}$** or **$\overline{B_1'B_2'}$**<br><br>
+\- or characterized by the *apparent angle* (in algebraic value or not) *substended by the object at nodal point* of the eye (direct vision) or nodal point of the thin imager : **$\alpha$** or **$\overline{\alpha}$**
+
+* Extended image [A'$_1$A'$_2$] **along the optical axis** :<br><br>
+characterized by the *algebraic size* (distance between its extremities) : **$\overline{A'_1A'_2}$** 
+
+##### Characterization of the imager action 
+
+The imager gives an image of an object. The **characterization of imager action** depends on *how the object and the image are characterized*.
+
+* *Object and image* both **characterized by their algebraic transverse sizes** :<br>
+**$\Longrightarrow$** *imager action* characterized by the **transverse magnification $M_T$**<br><br>
+**$M_T=\dfrac{image\:size}{object\:size}=\dfrac{\overline{A'B'}}{\overline{AB}}$**
+
+* *Object and image* both **characterized their apparent angles** :<br>
+**$\Longrightarrow$** *imager action* characterized by the **apparent magnification $M_A$**<br><br>
+**$M_A=\dfrac{image\:apparent\:angle}{object\:apparent\:angle}=\dfrac{\overline{\alpha'}}{\overline{\alpha}}$**<br><br>
+or $M_A=\pm\dfrac{\alpha'}{\alpha}$, with sign + when erect image, sign - when inverted image.<br><br>
+**apparent magnification** = **angular magnification**
+<br><br>
+_an apparent angle depends on distance from nodal point $\Longrightarrow$ more accurate definitions of apparent angles will be necessary (see chapter "optical instruments")_
+
+
+* *Object and image* both **characterized by their algebraic longitudinal sizes** :<br>
+**$\Longrightarrow$** *imager action* characterized by the **longitudinal magnification $M_L$**<br><br>
+**$M_L=\dfrac{image\:size}{object\:size}=\dfrac{\overline{A'_1A_2'}}{\overline{A_1A_2}}$**
+
+
+
+
+
+<!--
+![](objeto-fisico-imagen-virtual-haz-3-rayons-pantalla-bb.jpg)
+
+![](objeto-fisico-imagen-virtual-haz-3-rayos-bb.jpg)
+
+![](objeto-fisico-imagen-virtual-pantalla-bb.jpg)
+
+![](objeto-imagen-3-rayos-bb.jpg)
+
+![](objeto-imagen-haz-3-rayos-bb.jpg)
+
+![](plano-imagen-real-convergencia-bb.jpg)
+
+![](objeto-imagen-posicion-tamano-bb.jpg)
+
+![](plano-imagen-real-convergencia-pantalla-bb.jpg)
+
+![](plano-objeto-fisico-bb.jpg)
+
+![](plano-objeto-plano-imagen-real-pantalla-bb.jpg) -->
+
+
+
+
+
+
+
+
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----
-title: 'Modeling a mirror'
----
-
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+++ /dev/null
@@ -1,4 +0,0 @@
----
-title: 'Modelando un espejo'
----
-
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@@ -4,7 +4,7 @@ title: 'new course : beyond'
 
 ! *YOUR  CHALLENGE*   :   
 !
-! *CHALLENGE n°1* : The enigma of the cup of ice cream.
+! *CHALLENGE n°1* : The enigma of the cup of icecream.
 !
 ! ![](defi_glace_650.jpg)
 ! 
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diff --git a/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/01.plane-refracting-surface/topic.en.md b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/01.plane-refracting-surface/topic.en.md
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index 000000000..8bed5959a
--- /dev/null
+++ b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/01.plane-refracting-surface/topic.en.md
@@ -0,0 +1,4 @@
+---
+title: 'The spherical refracting surface '
+---
+
diff --git a/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/01.plane-refracting-surface/topic.es.md b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/01.plane-refracting-surface/topic.es.md
new file mode 100644
index 000000000..4e73ec027
--- /dev/null
+++ b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/01.plane-refracting-surface/topic.es.md
@@ -0,0 +1,6 @@
+---
+title: 'La superficie refractiva'
+---
+
+La superficie refractiva plana
+La superficie refractiva esferica
diff --git a/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/02.plane-refracting-surface/topic.fr.md b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/01.plane-refracting-surface/topic.fr.md
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diff --git a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/03.paraxial-optics-centered-systems/03.new-course-beyond/annex.en.md b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/02.mirror/03.new-course-beyond/annex.en.md
similarity index 54%
rename from 01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/03.paraxial-optics-centered-systems/03.new-course-beyond/annex.en.md
rename to 01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/02.mirror/03.new-course-beyond/annex.en.md
index 6ed0fb68e..e8c631f22 100644
--- a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/03.paraxial-optics-centered-systems/03.new-course-beyond/annex.en.md
+++ b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/02.mirror/03.new-course-beyond/annex.en.md
@@ -2,22 +2,33 @@
 title: 'new course : beyond'
 ---
 
-! *YOUR  CHALLENGE*  :  look at the picture, and think of the right answers to the following questions
+! *YOUR  CHALLENGE*  :  Your image in a mirror.
+! 
+!  ![](objet_image_defi_v2_300.jpg)
+!
+! We all have the opportunity to look at ourselves in a mirror. We see our own image. But perhaps it is interesting to take the time of interrogative observation, and then to test yourself.
 !
 ! _Do not look at the answer, take time to think, a few days if necessary. The time to build your mental representation of the phenomenon, to formulate it in words is important, a thousand times more important than the ephemeral instant where you read the fews words of the solution._
 !
 ! <details markdown=1>
 ! <summary>
-! INDICE "key word"
+! I choose.
 ! </summary>
-! diffusion
-! </details>
+! * Is my image simply located in my mind, or has it a location somewhere in space?
+! * If my image is localized, what is its position?
+! * If my image has a position, does it have a size? Is she taller or smaller than me?
+! * If my image has a position, and as it is made of light, can it illuminate a screen?
+! * If I retreat a little from the mirror, I see my image decreasing slightly. Why? Is the size of my image really smaller? Or does it keep its size and I'm the one who sees it smaller?
+! * Can I imagine to see more or less tall an image that has a well-defined size?
+!
+! These questions that I ask myself about my image, I have to ask them about an real object that I see in front of me.
 ! <details markdown=1>
 ! <summary>
 ! ANSWER
 ! </summary>
-! diffusion
+! To write.
 ! </details>
+! </summary>
 
 
 !!!! *DIFFICULT POINT* (contribute, or indicate a difficult point of understanding)
diff --git a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/03.paraxial-optics-centered-systems/03.new-course-beyond/annex.es.md b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/02.mirror/03.new-course-beyond/annex.es.md
similarity index 55%
rename from 01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/03.paraxial-optics-centered-systems/03.new-course-beyond/annex.es.md
rename to 01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/02.mirror/03.new-course-beyond/annex.es.md
index 790efacea..33413dc6c 100644
--- a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/03.paraxial-optics-centered-systems/03.new-course-beyond/annex.es.md
+++ b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/02.mirror/03.new-course-beyond/annex.es.md
@@ -2,23 +2,37 @@
 title: 'nuevo curso: más allá'
 ---
 
-! *TU DESAFÍO*  :  Mire la imagen y encuentre las respuestas correctas a las siguientes preguntas.
+! *TU DESAFÍO*  :  Tu imagen en un espejo.
+!
+! ![](objet_image_defi_v2_300.jpg)
+!
+! Todos tenemos la oportunidad de mirarnos en un espejo. Vemos nuestra propia imagen. Pero quizás sea interesante tomarse el tiempo de la observación interrogativa y luego probarse a sí mismo.
 !
 ! _No mire la respuesta, tómese el tiempo para pensar, unos días si es necesario. El momento de construir tu representación mental del fenómeno, de formular con palabras esta representación es importante, mil veces más importante que el momento efímero de leer las pocas palabras de la solución._
 !
 ! <details markdown=1>
 ! <summary>
-! ÍNDICE "palabra clave"
+! Elijo esto.
 ! </summary>
-! xxx
-! </details>
+! * ¿Mi imagen está simplemente ubicada en mi mente, o tiene una ubicación en el espacio?
+! * Si mi imagen está localizada, ¿cuál es su posición?
+! * Si mi imagen tiene una posición, ¿tiene un tamaño? ¿Es ella más alta o más pequeña que yo?
+! * Si mi imagen tiene una posición y, como está hecha de luz, ¿puede iluminar una pantalla?
+! * Si me retiro un poco del espejo, veo mi imagen más pequeña en el espejo. ¿Mi imagen es más pequeña? ¿O ella mantiene su tamaño y soy yo quien la ve más pequeña?
+! * ¿Puedo puedo ver una imagen más o menos grande, cuando esta imagen tiene un tamaño bien definido?
+!
+! Estas preguntas que me hago sobre mi imagen, tengo que hacerlas sobre un objeto real que veo frente a mí.
+!
 ! <details markdown=1>
 ! <summary>
 ! RESPUESTA
 ! </summary>
-! xxx
+! A escribir
+! </details>
+! </summary>
 ! </details>
 
+
 !!!! *PUNTO DIFÍCIL* (contribuya o indique los puntos difíciles que deben tomarse nuevamente)
 !!!!
 
diff --git a/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/01.mirror/03.new-course-beyond/annex.fr.md b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/02.mirror/03.new-course-beyond/annex.fr.md
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rename from 01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/01.mirror/03.new-course-beyond/annex.fr.md
rename to 01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/02.mirror/03.new-course-beyond/annex.fr.md
index 90d20c6e0..295a10b2a 100644
--- a/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/01.mirror/03.new-course-beyond/annex.fr.md
+++ b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/02.mirror/03.new-course-beyond/annex.fr.md
@@ -8,6 +8,9 @@ media_order: 'objet_image_defi_v2_650.jpg,objet_image_defi_v2_300.jpg'
 !  ![](objet_image_defi_v2_300.jpg)
 !
 ! Nous avons tous l'occasion de nous regarder dans un miroir. Nous y voyons notre propre image. Mais peut-être est-il intéressant de prendre le temps de l'observation interrogative, puis de se tester.
+!
+! _Ne regarde pas de suite la réponse, prends le temp de réfléchir, quelques jours si nécessaire. le temps pris à construire ta représentation mentale du phénomène, à le formaliser en mots, est important, des milliers de fois plus important que l'instant  éphémère où tu lis les quelques mots de la réponse_.
+!
 ! <details markdown=1>
 ! <summary>
 ! Je choisis ce défi
@@ -18,7 +21,9 @@ media_order: 'objet_image_defi_v2_650.jpg,objet_image_defi_v2_300.jpg'
 ! * Si mon image a une position, et comme elle est faite de lumière, peut-elle éclairer un écran?
 ! * Si je me recule un peu du miroir, je vois mon image plus petite dans le miroir. Est-ce mon image qui est plus petite? Ou bien garde-t-elle sa taille et c'est moi qui la voit plus petite ?
 ! * Puis-je imaginer voir plus ou moins grande une image qui aurait une taille bien définie?
+!
 ! Ces questions que je me pose sur mon image, je dois me les reposer sur un objet que je vois devant moi.
+!
 ! <details markdown=1>
 ! <summary>
 ! RÉPONSE
@@ -26,7 +31,6 @@ media_order: 'objet_image_defi_v2_650.jpg,objet_image_defi_v2_300.jpg'
 ! à détailler
 ! </details>
 ! </details>
-!
 
 
 !!!! *POINT  DIFFICILE* (contribuer, ou indiquer les points difficiles qui méritent d'être repris)
diff --git a/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/01.mirror/03.new-course-beyond/objet_image_defi_v2_300.jpg b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/02.mirror/03.new-course-beyond/objet_image_defi_v2_300.jpg
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+---
+title: 'The mirror'
+---
+
diff --git a/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/02.mirror/topic.es.md b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/02.mirror/topic.es.md
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+---
+title: 'El espejo'
+---
+
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----
-title: 'Spherical refracting surface '
----
-
-
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----
-title: 'Le dioptre '
----
-
-Dioptre sphérique,
-Dioptre plan
\ No newline at end of file
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@@ -1,4 +0,0 @@
----
-title: 'Simple modeling of the eye'
----
-
diff --git a/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/03.lens/01.new-course-main/textbook.en.md b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/03.lens/01.new-course-main/textbook.en.md
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+---
+title: 'new course : main'
+---
+
+### A lens
+
+An optical **lens** is a very common and *simple optical element*, *designed and made to focuse or disperse the light* by the use of the refraction phenomenon. Because requires to focus in a point image the light emitted or diffused by any object point
+
+<!--Optical lenses are the most important tools in optical design for controlling light. When optical designers talk about optical lenses, they are either referring to a single lens element or an assembly of lens elements (Figure 1). -->
+
+A lens is a **piece of transparent glass, quartz or plastic** that usually exhibits a *symmetry of revolution* about an axis<!-- or  being rotationally symmetrical-->. Its thickness, along its axis of symmetry, is small compared to the radius of its* two main surfaces* that are *polished and either or both curved*.
+
+The **curvation of each surface** can be charactérized as :
+* *convex* : the surface bulges outwards in the center.
+* *concave* : the surface curves inward in the center.
+* *plane* : the surface is plane.
+
+In the ideal case of lenses with negligeable thicknesses (_thin lenses_) or optical systems with negligeable length (_thin lenses, mirrors and refracting surfaces in contact_) :
+* The dioptric power (or vergence) $\delta$ of a *thin* lens is simply the sum of the dioptric powers  $\delta_{DS1}$ and $\delta_{DS2}$ of the two refracting surfaces $DS1$ and $DS2$ <!--interfaces? --> that compose the thin lens : $\delta=\delta_{DS1}+\delta_{DS2}$.
+* The dioptric power (vergence) $\delta$ of *thin lenses in contact or in contact with a mirror* is simply the sum of their individual dioptric powers $\delta_i$ : $\delta=\sum \delta_i$.
+
+In the more realistic and physical case, add a simple optical element of positive (respectively negative) dioptric power (or vergence) to an optical system will increase (resp. decrease) the total dioptric power of the system, so will increase the focalisation (resp. dispersion) of an incident light beam by the system.
+
+
+!! *Beyond* :
+!!
+!! Lentils gave lenses their name
+
+
+
+The final objective of lenses is to realize images, in the plane of a sensor to be recordered or to be seen with the naked eye. For this a 
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@@ -0,0 +1,144 @@
+---
+title: 'The thin lens'
+media_order: 'Const_lens_conv_point_AapresO.gif,lens-convergent-N2-en.jpeg,Const_lens_conv_point_AentreFO.gif,lens-convergent-N2-es.jpeg,lens-convergent-N2-fr.jpeg,Const_lens_conv_point_AavantF.gif,lens-divergent-N2-es.jpeg,lens-divergent-N2-fr.jpeg,lens-divergent-N2-en.jpeg,diverging-thin-lens-representation.jpeg,converging-thin-lens-representation.jpeg'
+---
+
+### What is a lens ?
+
+#### Objective
+
+* initial : to **focuse or disperse the light**.
+* ultimate : to **realize images**, alone or as part of optical instruments.
+
+#### Physical principle
+
+* **uses the refractive phenomenon**, described by the Snell-Descartes' law.
+
+#### Constitution
+
+* Piece of **glass, quartz, plastic** (for visible and near infrared and UV).
+* **Rotationally symmetrical**.
+* **2 polished surfaces** perpendicular to its axis of symmetry, **either or both curved** (and most often spherical).
+
+<!--image to build : a thin lens-->
+
+#### Interest in optics : thin lenses
+
+* **Thin lens** : *thickness << diameter*
+* Thins lens : **most important simple optical element** that is *used alone or combined in serie in most optical instruments* : magnifying glasses, microscopes, tele and macro objectives, camera, refracting telescopes.
+
+<!--image to build N1 ou N2 : a composition :
+upper medium : a unic thin lens
+upper part towards utilization of a unique lens : magnigfying glass and eyeglasses
+lower medium : small serie of centered naked lenses
+lower part toward utilization of combined lenses : macroscope, camera (apparatus and objective of a cellular), refracting telescope, teleopbjective-->
+
+### Modeling a thin lens surrounded by air, gaz or vaccum.
+
+#### Why modeling ?
+
+* To **understand, calculate and predict images** of objects given by thin lenses
+
+<!--picture when we see the object, the lens and the image-->
+
+##### Why  surrounded by air, gaz or vaccum?
+
+* **In most optical instruments**, lenses are *surrounding by air*.
+* **air, gaz and vaccum** have refractive index values in the range "$1.000\pm0.001$, and can be approximated by *$n_{air}=n_{gaz}=n_{vaccum}=1$*<br>
+$\Longrightarrow$ same optical behavior in air, gaz and vacuum.
+
+#### Types and characterization of thin lenses
+
+**Convergent** = **converging** = **convexe** = **positive** lenses
+
+![](lens-convergent-N2-en.jpeg)
+
+* Characterized by :<br>
+\- **Focal lenght** (usually in cm) always >0 *+* adjective "**converging**"<br>
+&nbsp;&nbsp;or<br>
+\- Its **image focal length** $f'$ (in *algebraic value*, usually in cm), that is *positive $f'>0$*.<br>
+&nbsp;&nbsp;or<br>
+\- Its **vergence** $V$ (in ophtalmology) that is *positive  $V>0$*,<br>
+with $V (\delta)=\dfrac{1}{f'(m)}$ ($f'$ being expresssed in m "meter" and $V$ in $\delta$ "dioptre", so $\delta=m^{-1}$).<br>
+ 
+**Divergent** = **diverging** = **concave ** = **negative**  lenses
+
+![](lens-divergent-N2-en.jpeg)
+
+* Characterized by :<br>
+\- **Focal lenght** (usually in cm) always <0 *+* adjective "**diverging**"<br>
+&nbsp;&nbsp;or<br>
+\- Its **image focal length** $f'$ (in *algebraic value*, usually in cm), that is *negative $f'<0$*.<br>
+&nbsp;&nbsp;or<br>
+\- Its **vergence** $V$ (in ophtalmology) that is *negative  $V<0$*,<br>
+with $V (\delta)=\dfrac{1}{f'(m)}$ ($f'$ being expresssed in m "meter" and $V$ in $\delta$ "dioptre", so $\delta=m^{-1}$).<br>
+
+<!-- suppressed
+#### What physical framework, model and technics ?
+
+* _Framework : Geometrical Optics = Light rays optics $\longrightarrow$ foothills stage_.
+
+* _Model : paraxial model = gaussian model $\longrightarrow$ foothills stage_.
+
+* Model splits in *two different technics (but equivalent)* :<br> **graphical modeling** AND **analytical modeling**
+
+* *Differences between model predictions and experimental observations* : ** optical aberrations** (_under control, minimized and negligeable in optical instruments_).
+-->
+
+### Analytical modeling
+
+(_for thin lens surrounded by air, gaz or vaccum_)
+
+##### Thin lens equation
+**$\dfrac{1}{\overline{OA'}}-\dfrac{1}{\overline{OA}}=V=-\dfrac{1}{\overline{OF}}=\dfrac{1}{\overline{OF'}}$**
+
+##### Transverse magnification expression 
+**$M_{T-thinlens}=\dfrac{\overline{OA'}}{\overline{OA}}$** 
+
+
+### Graphical modeling
+
+#### Thin lens representation
+
+* **optical axis** = *revolution axis* of the lens, positively *oriented* in the direction of propagation of the light (_from the object towards the lens_).
+
+* **thins lens representation** :<br><br>
+\- *line segment*, perpendicular to optical axis, centered on the axis with symbolic *indication of the lens shape* at its extremities (_convexe or concave_).<br><br>
+\- **S = C = O** : vertex S = nodal point C = center O of the thin lens $\Longrightarrow$ is used point O.<br><br>
+\- *point O*, intersection of the line segment with optical axis.<br><br>
+\- *object focal point F* and *image focal point F'*, positioned on the optical axis symmetrically with respect to the point O ($f=-f'$) at algebraic distances $\overline{OF}=f$ and $\overline{OF'}=f'$.<br><br> 
+\- *object focal plane (P)* and *image focal plane (P')*, planes perpendicular to the optical axis at respectively points $F$ and $F'$.
+
+![](converging-thin-lens-representation.jpeg)<br>
+_Converging thin lens representation : $\overline{OF}<0$ , $\overline{OF'}>0$ and $|\overline{OF}|=|\overline{OF'}|$_
+
+ ![](diverging-thin-lens-representation.jpeg)<br>
+ _Divverging thin lens representation : $\overline{OF}>0$ , $\overline{OF'}<0$ and $|\overline{OF}|=|\overline{OF'}|$_
+ 
+#### Determining conjugate points :
+
+##### Converging thin lens
+
+**Towards geogebra animations** :<br>
+\- Graphical construction<br>
+[Click here for geogebra animation](https://www.geogebra.org/material/iframe/id/zqwazusz)<br>
+\- Graphical construction and light pencils <br>
+[Click here for geogebra animation](https://www.geogebra.org/material/iframe/id/wkrw5qgm)<br>
+\- Graphical construction and transverse magnification<br>
+[Click here for geogebra animation](https://www.geogebra.org/material/iframe/id/xwbwedft)<br>
+
+* **Point source located between &infin; et F**
+
+![](Const_lens_conv_point_AavantF.gif)
+
+* **Point source located between F et O**
+
+![](Const_lens_conv_point_AentreFO.gif)
+
+* **Virtual object point** (will be seen at level foothills, to remove from here).
+
+![](Const_lens_conv_point_AapresO.gif)
+
+##### Diverging thin lens 
+
+(to be implemented)
diff --git a/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/03.lens/02.new-course-overview/cheatsheet.es.md b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/03.lens/02.new-course-overview/cheatsheet.es.md
new file mode 100644
index 000000000..abffc6cae
--- /dev/null
+++ b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/03.lens/02.new-course-overview/cheatsheet.es.md
@@ -0,0 +1,14 @@
+---
+title: ' La lente delgada'
+media_order: 'lens-divergent-N2-es.jpeg,lens-convergent-N2-en.jpeg,lens-convergent-N2-es.jpeg,lens-convergent-N2-fr.jpeg,lens-divergent-N2-en.jpeg,lens-divergent-N2-fr.jpeg'
+---
+
+### La vergencia de una lente
+
+### Los diferentes tipos de lentes
+
+![](lens-convergent-N2-es.jpeg)
+
+![](lens-divergent-N2-es.jpeg)
+
+### Modelado de una lente.
\ No newline at end of file
diff --git a/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/03.lens/02.new-course-overview/cheatsheet.fr.md b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/03.lens/02.new-course-overview/cheatsheet.fr.md
new file mode 100644
index 000000000..bf5aeb9fd
--- /dev/null
+++ b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/03.lens/02.new-course-overview/cheatsheet.fr.md
@@ -0,0 +1,14 @@
+---
+title: 'La lentille mince'
+media_order: 'lens-convergent-N2-en.jpeg,lens-divergent-N2-fr.jpeg,lens-convergent-N2-es.jpeg,lens-convergent-N2-fr.jpeg,lens-divergent-N2-en.jpeg,lens-divergent-N2-es.jpeg'
+---
+
+### La vergence d'une lentille
+
+### Les différents types de lentilles
+
+![](lens-convergent-N2-fr.jpeg)
+
+![](lens-divergent-N2-fr.jpeg)
+
+### Modélisation d'une lentille mince
\ No newline at end of file
diff --git a/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/03.lens/02.new-course-overview/converging-thin-lens-representation.jpeg b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/03.lens/02.new-course-overview/converging-thin-lens-representation.jpeg
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diff --git a/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/03.lens/03.new-course-beyond/annex.en.md b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/03.lens/03.new-course-beyond/annex.en.md
new file mode 100644
index 000000000..0f0f2053c
--- /dev/null
+++ b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/03.lens/03.new-course-beyond/annex.en.md
@@ -0,0 +1,124 @@
+---
+title: 'new course : beyond'
+---
+
+! *YOUR  CHALLENGE*   :   
+!
+! *CHALLENGE n°1* : xxx
+!
+! An image
+! 
+! The question
+!
+! <details markdown=1>
+! <summary>
+! I choose this challenge.
+! </summary>
+! Information : xxx
+!
+! _Don’t look at the answer, take some time to think, a few days if necessary. The time to build your mental representation of the phenomenon, to formulate it in words is important, a thousand times more important than the ephemeral moment of reading the few words of the solution._
+! 
+! <details markdown=1>
+! <summary>
+! CLUE 1 : keywords
+! </summary>
+! xxx
+! </details>
+! <details markdown=1>
+! <summary>
+! CLUE 2  : trail
+! </summary>
+! xxx
+! </details>
+! <details markdown=1>
+! <summary>
+! ANSWER
+! </summary>
+! to write
+! </details>
+! </details>
+!
+! -----------
+!
+! *CHALLENGE n°2* : xxx
+!
+! An image
+!
+! The question
+!
+! <details markdown=1>
+! <summary>
+! I choose this challenge
+! </summary>
+! Information : xxx
+!
+! _Don’t look at the answer, take some time to think, a few days if necessary. The time to build your mental representation of the phenomenon, to formulate it in words is important, a thousand times more important than the ephemeral moment of reading the few words of the solution._
+!
+! <details markdown=1>
+! <summary>
+! CLUE 1 : keywords
+! </summary>
+! xxx
+! </details>
+! <details markdown=1>
+! <summary>
+! CLUE 2 : trail
+! </summary>
+! xxx
+! </details>
+! <details markdown=1>
+! <summary>
+! ANSWER
+! </summary>
+! to write
+! </details>
+! </details>
+!
+! -----------
+!
+! *CHALLENGE n°3* : A multi touch screen technology (challenge under construction)
+!
+! -----------
+!
+! *CHALLENGE n°4* : Luminous fountain (challenge under construction)
+!
+
+
+!!!! *DIFFICULT POINT* (contribute, or indicate a difficult point of understanding)
+!!!!
+
+!! *BEYOND* (to contribute)
+!!
+
+!! *CULTURAL POINT* (contributor)
+!!
+
+!!! *DO  YOU  MASTER ?*
+!!! <details markdown=1>
+!!! <Summary>
+!!! Describe the test
+!!! </summary>
+!!! The text of the test (and its images, figures, audio, video, etc ...)
+!!!
+!!! Question text
+!!! <details markdown=1>
+!!! <summary>
+!!! Answer choice 1
+!!! </summary>
+!!! Text if this answer 1 is chosen
+!!! </details>
+!!! <details markdown=1>
+!!! <summary>
+!!! Answer choice 2
+!!! </summary>
+!!! Text if this answer 2 is chosen
+!!! </details>
+!!! <!--possibility to add answers, questions-->
+!!! -----
+!!! <details markdown=1>
+!!! <summary>
+!!! Complete solution.
+!!! </summary>
+!!! text of the solution
+!!! </details>
+
diff --git a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.physical-system-versus-optical-system/03.new-course-beyond/annex.es.md b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/03.lens/03.new-course-beyond/annex.es.md
similarity index 50%
rename from 01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.physical-system-versus-optical-system/03.new-course-beyond/annex.es.md
rename to 01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/03.lens/03.new-course-beyond/annex.es.md
index 790efacea..5e4c8fbed 100644
--- a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.physical-system-versus-optical-system/03.new-course-beyond/annex.es.md
+++ b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/03.lens/03.new-course-beyond/annex.es.md
@@ -2,13 +2,30 @@
 title: 'nuevo curso: más allá'
 ---
 
-! *TU DESAFÍO*  :  Mire la imagen y encuentre las respuestas correctas a las siguientes preguntas.
+! *TU DESAFÍO*  
+!
+! *DESAFÍO n°1* : xxx
+!
+! Una imagen
+! 
+! La cuestión
+! <details markdown=1>
+! <summary>
+! Elijo este desafío.
+! </summary>
+! Datos: xxx
 !
 ! _No mire la respuesta, tómese el tiempo para pensar, unos días si es necesario. El momento de construir tu representación mental del fenómeno, de formular con palabras esta representación es importante, mil veces más importante que el momento efímero de leer las pocas palabras de la solución._
 !
 ! <details markdown=1>
 ! <summary>
-! ÍNDICE "palabra clave"
+! ÍNDICE 1: palabras clave
+! </summary>
+! xxx
+! </details>
+! <details markdown=1>
+! <summary>
+! ÍNDICE 2: palabras clave
 ! </summary>
 ! xxx
 ! </details>
@@ -16,8 +33,54 @@ title: 'nuevo curso: más allá'
 ! <summary>
 ! RESPUESTA
 ! </summary>
+! a escribir
+! </details>
+! </details>
+!
+! -----------
+!
+! *DESAFÍO n ° 2* : xxx
+!
+! Una imagen
+!
+! La cuestión
+!
+! <details markdown=1>
+! <summary>
+! Elijo este desafío.
+! </summary>
+! Datos: xxx
+!
+! _No mire la respuesta, tómese el tiempo para pensar, unos días si es necesario. El momento de construir tu representación mental del fenómeno, de formular con palabras esta representación es importante, mil veces más importante que el momento efímero de leer las pocas palabras de la solución._
+!
+! <details markdown=1>
+! <summary>
+! ÍNDICE 1: palabras clave
+! </summary>
 ! xxx
 ! </details>
+! <details markdown=1>
+! <summary>
+! ÍNDICE 2: palabras clave
+! </summary>
+! xxx
+! </details>
+! <details markdown=1>
+! <summary>
+! RESPUESTA
+! </summary>
+! a escribir
+! </details>
+! </details>
+!
+! -----------
+!
+! *DESAFÍO n°3* : Una tecnología de pantalla táctil multi-puntos de contacto (desafío en construcción)
+!
+! -----------
+!
+! *DESAFÍO n°4* : Fuente luminosa (desafío en construcción)
+!
 
 !!!! *PUNTO DIFÍCIL* (contribuya o indique los puntos difíciles que deben tomarse nuevamente)
 !!!!
@@ -28,7 +91,7 @@ title: 'nuevo curso: más allá'
 !! *ESCAPE CULTURAL* (contribuir)
 !!
 
-!!! *MAÎTRISES-TU ?*
+!!! *DOMINAS ?*
 !!! <details markdown = 1>
 !!! <summary>
 !!! Describe la prueba
diff --git a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.physical-system-versus-optical-system/03.new-course-beyond/annex.fr.md b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/03.lens/03.new-course-beyond/annex.fr.md
similarity index 52%
rename from 01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.physical-system-versus-optical-system/03.new-course-beyond/annex.fr.md
rename to 01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/03.lens/03.new-course-beyond/annex.fr.md
index 0619fc5f7..4a96aed11 100644
--- a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.physical-system-versus-optical-system/03.new-course-beyond/annex.fr.md
+++ b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/03.lens/03.new-course-beyond/annex.fr.md
@@ -1,14 +1,33 @@
 ---
-title: 'nouveau cours : au-delà'
+title: 'Au-delà'
+media_order: 'objet_image_defi_v2_650.jpg,objet_image_defi_v2_300.jpg'
 ---
 
 ! *TON  DEFI*  :   
 !
+! *DEFI n°1* : xxx
+!
+! Image explicative
+! 
+! La question
+!
+! <details markdown=1>
+! <summary>
+! Je choisis ce défi
+! </summary>
+! xxxx
+!
 ! _Ne regarde pas la réponse, prends le temps de réfléchir, quelques jours si nécessaire. Le temps de construire ta représentation mentale du phénomène, de formuler en mots cette réflexion est important, un millier de fois plus important que l'instant éphémère de la lecture des quelques mots de la solution._
 !
 ! <details markdown=1>
 ! <summary>
-! INDICE "mot clé"
+! INDICE 1 "mots clé"
+! </summary>
+! xxx
+! </details>
+! <details markdown=1>
+! <summary>
+! INDICE 2 "indication"
 ! </summary>
 ! xxx
 ! </details>
@@ -16,8 +35,54 @@ title: 'nouveau cours : au-delà'
 ! <summary>
 ! RÉPONSE
 ! </summary>
+! à détailler
+! </details>
+! </details>
+!
+! -----------
+!
+! *DEFI n°2* : xxx
+!
+! L'image
+!
+! La question
+!
+! <details markdown=1>
+! <summary>
+! Je choisis ce défi
+! </summary>
+! Données : xxxx.
+!
+! _Ne regarde pas la réponse, prends le temps de réfléchir, quelques jours si nécessaire. Le temps de construire ta représentation mentale du phénomène, de formuler en mots cette réflexion est important, un millier de fois plus important que l'instant éphémère de la lecture des quelques mots de la solution._
+!
+! <details markdown=1>
+! <summary>
+! INDICE 1 "mots clé"
+! </summary>
 ! xxx
 ! </details>
+! <details markdown=1>
+! <summary>
+! INDICE 2 "indication"
+! </summary>
+! xxx
+! </details>
+! <details markdown=1>
+! <summary>
+! RÉPONSE
+! </summary>
+! à détailler
+! </details>
+! </details>
+!
+! -----------
+!
+! *DEFI n°3* : xxx
+!
+! -----------
+!
+! *DEFI n°4* : xxx
+!
 
 !!!! *POINT  DIFFICILE* (contribuer, ou indiquer les points difficiles qui méritent d'être repris)
 !!!!
diff --git a/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/04.new-course-parallel-1/default.en.md b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/03.lens/04.new-course-parallel-1/default.en.md
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diff --git a/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/03.lens/textbook.en.md b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/03.lens/textbook.en.md
index 223d879fc..18890e2a5 100644
--- a/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/03.lens/textbook.en.md
+++ b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/03.lens/textbook.en.md
@@ -1,4 +1,4 @@
 ---
-title: 'Modeling a thin lens'
+title: 'The thin lens'
 ---
 
diff --git a/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/03.lens/textbook.es.md b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/03.lens/textbook.es.md
new file mode 100644
index 000000000..4bda1097a
--- /dev/null
+++ b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/03.lens/textbook.es.md
@@ -0,0 +1,4 @@
+---
+title: 'La lente delgada '
+---
+
diff --git a/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/03.lens/textbook.fr.md b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/03.lens/textbook.fr.md
new file mode 100644
index 000000000..a69dee1b1
--- /dev/null
+++ b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/03.lens/textbook.fr.md
@@ -0,0 +1,4 @@
+---
+title: 'Le lentille mince'
+---
+
diff --git a/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/03.lens/topic.es.md b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/03.lens/topic.es.md
index bf8057520..a2306078f 100644
--- a/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/03.lens/topic.es.md
+++ b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/03.lens/topic.es.md
@@ -1,4 +1,4 @@
 ---
-title: 'Modelando una lente delgada'
+title: 'Modelando un espejo'
 ---
 
diff --git a/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/03.lens/topic.fr.md b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/03.lens/topic.fr.md
index d08fa960f..284a0b0ee 100644
--- a/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/03.lens/topic.fr.md
+++ b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/03.lens/topic.fr.md
@@ -1,4 +1,4 @@
 ---
-title: 'Modélisation d''une lentille mince'
+title: 'Le miroir'
 ---
 
diff --git a/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/04.eye/textbook.en.md b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/04.eye/textbook.en.md
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--- /dev/null
+++ b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/04.eye/textbook.en.md
@@ -0,0 +1,4 @@
+---
+title: 'The eye'
+---
+
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index 7118f9fbd..07b26c229 100644
--- a/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/topic.es.md
+++ b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/topic.es.md
@@ -1,5 +1,5 @@
 ---
-title: Optique
+title: Óptico
 slug: optics
 ---
 
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index aacd76fe5..e0a1eeb28 100644
--- a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/02.geometrical-optics-foundings/03.optical_path/01.optical-path-main/textbook.en.md
+++ b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/02.geometrical-optics-foundings/03.optical_path/01.optical-path-main/textbook.en.md
@@ -2,28 +2,34 @@
 title: 'The optical path T'
 ---
 
+TO  TRANSLATE
+
 #####Le chemin optique
 
 Lorsque la lumière se déplace sur des trajectoires traversant des milieux d'indices réfraction différents, les différents temps de parcours ne sont pas égaux aux longueurs correspondantes multipliées par un nombre réel unique, indépendant des trajectoires. Cela reste vrai sur la trajectoire unique d'un rayon de lumière traversant plusieurs milieux : pour une même longueur considérée le long de la trajectoire, le temps de parcours pourra être différent selon la portion de trajectoire sur laquelle la longueur est prise. Je peux résumer cela d'une phrase :
 
-<ins>Sur l'ensemble des cas,</ins><strong> le temps de parcours n'est pas proportionnel à la distance parcourue.</strong>
+*Sur l'ensemble des cas*, **le temps de parcours n'est pas proportionnel à la distance parcourue.**
+
+Or la grandeur physique importante associée à un parcours entre deux points de l'espace, pour savoir si la lumière choisira ce parcours plutôt q'un autre, sera le temps de parcours. Cependant lorsque je visualise mentalement une trajectoire, je vois une ligne à laquelle j'associe intuitivement une longueur. *Comment travailler mathématiquement avec une grandeur physique homogène à une longueur, mais qui aurait les mêmes propriétés que le temps de parcours pour décrire la propagation de la lumière *? Une telle grandeur a été définie en physique, et est grandement utilisée en optique géométrique, optique ondulatoire, électromagnétisme, et elle est nommée " **chemin optique** noté usuellement " **$\delta_o$**".
 
-Or la grandeur physique importante associée à un parcours entre deux points de l'espace, pour savoir si la lumière choisira ce parcours plutôt q'un autre, sera le temps de parcours. Cependant lorsque je visualise mentalement une trajectoire, je vois une ligne à laquelle j'associe intuitivement une longueur. <ins>Comment travailler mathématiquement avec une grandeur physique homogène à une longueur, mais qui aurait les mêmes propriétés que le temps de parcours pour décrire la propagation de la lumière </ins>? Une telle grandeur a été définie en physique, et est grandement utilisée en optique géométrique, optique ondulatoire, électromagnétisme, et elle est nommée "<strong>chemin optique</strong> noté usuellement "<strong> $\delta_o$</strong>".
+Le chemin optique $\delta_o$ d'un parcours donné $\Gamma_o$ entre deux points A et B de l'espace est *homogène à une longueur*. Son **unité (S.I.)** (son unité dans le Système International d'unités) est donc le "*mètre*".
 
-Le chemin optique $\delta_o$ d'un parcours donné $\Gamma_o$ entre deux points A et B de l'espace est <ins>homogène à une longueur</ins>. Son <strong>unité (S.I.)</strong> (son unité dans le Système International d'unités) est donc le "<ins>mètre</ins>".
+Pour tout segment de droite pris entre deux points infiniment proche dans l'espace, son chemin optique infinitésimal (ou élémentaire) **mathrm{d}\delta$** est égal à sa *longueur euclidienne $\mathrm{d}s$ multipliée par la valeur de l'indice de réfraction $n$* moyennée sur le segment infinitésimal considéré :
 
-Pour tout segment de droite pris entre deux points infiniment proche dans l'espace, son chemin optique infinitésimal (ou élémentaire) <strong>$\mathrm{d}\delta$</strong> est égal à sa <ins>longueur euclidienne $\mathrm{d}s$ multipliée par la valeur de l'indice de réfraction $n$</ins> moyennée sur le segment infinitésimal considéré :
-<strong>$\mathrm{d}\delta\;=\;n\times \mathrm{d}s$</strong>
+**$\mathrm{d}\delta\;=\;n\times \mathrm{d}s$**
 
 Le chemin optique $\delta$ d'un parcours donné $\Gamma_o$ entre deux point de l'espace est simplement la somme des chemins optiques infinitésimaux intégrés sur ce parcours :
-<strong>$\delta = \int_{\Gamma_o}\mathrm{d}\delta= \int_{\Gamma_o}n\cdot \mathrm{d}s$</strong>
 
-Quelques soient deux points donnés A et B de l'espace, et quelque soit le parcours considéré entre ces deux points, le <strong>chemin optique</strong> sera <ins>toujours égal au temps de parcours de la lumière sur ce parcours divisé par la vitesse de la lumière dans le vide $c$ </ins>qui est une constante universelle de la nature :
-<strong>$\mathrm{d}\delta\;=\;\frac{ds}{c}$
-$\delta = \int_{S_{AB}}n\cdot\mathrm{d}s\;=\;\int_{S_{AB}}\frac{c}{v}\cdot\mathrm{d}s$
-$\hspace{1cm}= c\;\int_{S_{AB}}\frac{\mathrm{d}s}{v} =\;c\;\tau$</strong>
+**$\delta = \int_{\Gamma_o}\mathrm{d}\delta= \int_{\Gamma_o}n\cdot \mathrm{d}s$**
 
+Quelques soient deux points donnés A et B de l'espace, et quelque soit le parcours considéré entre ces deux points, le **chemin optique** sera *toujours égal au temps de parcours de la lumière sur ce parcours divisé par la vitesse de la lumière dans le vide $c$ *qui est une constante universelle de la nature :
+
+**$\mathrm{d}\delta\;=\;\frac{ds}{c}$
+$\delta = \int_{S_{AB}}n\cdot\mathrm{d}s\;=\;\int_{S_{AB}}\frac{c}{v}\cdot\mathrm{d}s$
+$\hspace{1cm}= c\;\int_{S_{AB}}\frac{\mathrm{d}s}{v} =\;c\;\tau$**
 
-<br>
-<ul class="exemple">Je peux maintenant considérer un rayon lumineux se propageant d'un point A à un point B, et lui imposer au cours de sa trajectoire entre A et B d'interagir avec un système optique. Je peux ensuite considérer l'ensemble des chemins possibles (ils sont en nombre infini en optique) entre A et B, et considérer une application f qui à chaque chemin de cet ensemble associe une grandeur physique particulière. En optique géométrique, les deux grandeurs physiques intéressantes sont le temps de parcours et le chemin optique. La question est :<br>
-Le trajet réellement suivi par la lumière dans chaque cas correspond-t-il à un chemin défini par un point particulier de la fonction f ?. Les points particuliers qui vont m'intéresser en optique géométrique sont appelés en mathématiques les points stationnaires.</ul>
+!!! Je peux maintenant considérer un rayon lumineux se propageant d'un point A à un point B, et lui imposer au cours de sa trajectoire entre A et B d'interagir avec un système optique. Je peux ensuite considérer l'ensemble des chemins possibles (ils sont en nombre infini en optique) entre A et B, et considérer une application f qui à chaque chemin de cet ensemble associe une grandeur physique particulière. En optique géométrique, les deux grandeurs physiques intéressantes sont le temps de parcours et le chemin optique. La question est :
+!!!
+!!! Le trajet réellement suivi par la lumière dans chaque cas correspond-t-il à un chemin défini par un point particulier de la fonction f ?
+!!!
+!!! Les points particuliers qui vont m'intéresser en optique géométrique sont appelés en mathématiques les points stationnaires.
diff --git a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/02.geometrical-optics-foundings/03.optical_path/01.optical-path-main/textbook.es.md b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/02.geometrical-optics-foundings/03.optical_path/01.optical-path-main/textbook.es.md
index ab6376544..60601f122 100644
--- a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/02.geometrical-optics-foundings/03.optical_path/01.optical-path-main/textbook.es.md
+++ b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/02.geometrical-optics-foundings/03.optical_path/01.optical-path-main/textbook.es.md
@@ -2,28 +2,34 @@
 title: 'El camino óptico T'
 ---
 
+A  TRADUCIR
+
 #####Le chemin optique
 
 Lorsque la lumière se déplace sur des trajectoires traversant des milieux d'indices réfraction différents, les différents temps de parcours ne sont pas égaux aux longueurs correspondantes multipliées par un nombre réel unique, indépendant des trajectoires. Cela reste vrai sur la trajectoire unique d'un rayon de lumière traversant plusieurs milieux : pour une même longueur considérée le long de la trajectoire, le temps de parcours pourra être différent selon la portion de trajectoire sur laquelle la longueur est prise. Je peux résumer cela d'une phrase :
 
-<ins>Sur l'ensemble des cas,</ins><strong> le temps de parcours n'est pas proportionnel à la distance parcourue.</strong>
+*Sur l'ensemble des cas*, **le temps de parcours n'est pas proportionnel à la distance parcourue.**
+
+Or la grandeur physique importante associée à un parcours entre deux points de l'espace, pour savoir si la lumière choisira ce parcours plutôt q'un autre, sera le temps de parcours. Cependant lorsque je visualise mentalement une trajectoire, je vois une ligne à laquelle j'associe intuitivement une longueur. *Comment travailler mathématiquement avec une grandeur physique homogène à une longueur, mais qui aurait les mêmes propriétés que le temps de parcours pour décrire la propagation de la lumière *? Une telle grandeur a été définie en physique, et est grandement utilisée en optique géométrique, optique ondulatoire, électromagnétisme, et elle est nommée " **chemin optique** noté usuellement " **$\delta_o$**".
 
-Or la grandeur physique importante associée à un parcours entre deux points de l'espace, pour savoir si la lumière choisira ce parcours plutôt q'un autre, sera le temps de parcours. Cependant lorsque je visualise mentalement une trajectoire, je vois une ligne à laquelle j'associe intuitivement une longueur. <ins>Comment travailler mathématiquement avec une grandeur physique homogène à une longueur, mais qui aurait les mêmes propriétés que le temps de parcours pour décrire la propagation de la lumière </ins>? Une telle grandeur a été définie en physique, et est grandement utilisée en optique géométrique, optique ondulatoire, électromagnétisme, et elle est nommée "<strong>chemin optique</strong> noté usuellement "<strong> $\delta_o$</strong>".
+Le chemin optique $\delta_o$ d'un parcours donné $\Gamma_o$ entre deux points A et B de l'espace est *homogène à une longueur*. Son **unité (S.I.)** (son unité dans le Système International d'unités) est donc le "*mètre*".
 
-Le chemin optique $\delta_o$ d'un parcours donné $\Gamma_o$ entre deux points A et B de l'espace est <ins>homogène à une longueur</ins>. Son <strong>unité (S.I.)</strong> (son unité dans le Système International d'unités) est donc le "<ins>mètre</ins>".
+Pour tout segment de droite pris entre deux points infiniment proche dans l'espace, son chemin optique infinitésimal (ou élémentaire) **mathrm{d}\delta$** est égal à sa *longueur euclidienne $\mathrm{d}s$ multipliée par la valeur de l'indice de réfraction $n$* moyennée sur le segment infinitésimal considéré :
 
-Pour tout segment de droite pris entre deux points infiniment proche dans l'espace, son chemin optique infinitésimal (ou élémentaire) <strong>$\mathrm{d}\delta$</strong> est égal à sa <ins>longueur euclidienne $\mathrm{d}s$ multipliée par la valeur de l'indice de réfraction $n$</ins> moyennée sur le segment infinitésimal considéré :
-<strong>$\mathrm{d}\delta\;=\;n\times \mathrm{d}s$</strong>
+**$\mathrm{d}\delta\;=\;n\times \mathrm{d}s$**
 
 Le chemin optique $\delta$ d'un parcours donné $\Gamma_o$ entre deux point de l'espace est simplement la somme des chemins optiques infinitésimaux intégrés sur ce parcours :
-<strong>$\delta = \int_{\Gamma_o}\mathrm{d}\delta= \int_{\Gamma_o}n\cdot \mathrm{d}s$</strong>
 
-Quelques soient deux points donnés A et B de l'espace, et quelque soit le parcours considéré entre ces deux points, le <strong>chemin optique</strong> sera <ins>toujours égal au temps de parcours de la lumière sur ce parcours divisé par la vitesse de la lumière dans le vide $c$ </ins>qui est une constante universelle de la nature :
-<strong>$\mathrm{d}\delta\;=\;\frac{ds}{c}$
-$\delta = \int_{S_{AB}}n\cdot\mathrm{d}s\;=\;\int_{S_{AB}}\frac{c}{v}\cdot\mathrm{d}s$
-$\hspace{1cm}= c\;\int_{S_{AB}}\frac{\mathrm{d}s}{v} =\;c\;\tau$</strong>
+**$\delta = \int_{\Gamma_o}\mathrm{d}\delta= \int_{\Gamma_o}n\cdot \mathrm{d}s$**
 
+Quelques soient deux points donnés A et B de l'espace, et quelque soit le parcours considéré entre ces deux points, le **chemin optique** sera *toujours égal au temps de parcours de la lumière sur ce parcours divisé par la vitesse de la lumière dans le vide $c$ *qui est une constante universelle de la nature :
+
+**$\mathrm{d}\delta\;=\;\frac{ds}{c}$
+$\delta = \int_{S_{AB}}n\cdot\mathrm{d}s\;=\;\int_{S_{AB}}\frac{c}{v}\cdot\mathrm{d}s$
+$\hspace{1cm}= c\;\int_{S_{AB}}\frac{\mathrm{d}s}{v} =\;c\;\tau$**
 
-<br>
-<ul class="exemple">Je peux maintenant considérer un rayon lumineux se propageant d'un point A à un point B, et lui imposer au cours de sa trajectoire entre A et B d'interagir avec un système optique. Je peux ensuite considérer l'ensemble des chemins possibles (ils sont en nombre infini en optique) entre A et B, et considérer une application f qui à chaque chemin de cet ensemble associe une grandeur physique particulière. En optique géométrique, les deux grandeurs physiques intéressantes sont le temps de parcours et le chemin optique. La question est :<br>
-Le trajet réellement suivi par la lumière dans chaque cas correspond-t-il à un chemin défini par un point particulier de la fonction f ?. Les points particuliers qui vont m'intéresser en optique géométrique sont appelés en mathématiques les points stationnaires.</ul>
+!!! Je peux maintenant considérer un rayon lumineux se propageant d'un point A à un point B, et lui imposer au cours de sa trajectoire entre A et B d'interagir avec un système optique. Je peux ensuite considérer l'ensemble des chemins possibles (ils sont en nombre infini en optique) entre A et B, et considérer une application f qui à chaque chemin de cet ensemble associe une grandeur physique particulière. En optique géométrique, les deux grandeurs physiques intéressantes sont le temps de parcours et le chemin optique. La question est :
+!!!
+!!! Le trajet réellement suivi par la lumière dans chaque cas correspond-t-il à un chemin défini par un point particulier de la fonction f ?
+!!!
+!!! Les points particuliers qui vont m'intéresser en optique géométrique sont appelés en mathématiques les points stationnaires.
diff --git a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/02.geometrical-optics-foundings/03.optical_path/01.optical-path-main/textbook.fr.md b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/02.geometrical-optics-foundings/03.optical_path/01.optical-path-main/textbook.fr.md
index bca54b0af..5d4bd961b 100644
--- a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/02.geometrical-optics-foundings/03.optical_path/01.optical-path-main/textbook.fr.md
+++ b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/02.geometrical-optics-foundings/03.optical_path/01.optical-path-main/textbook.fr.md
@@ -6,24 +6,28 @@ title: 'Le chemin optique T'
 
 Lorsque la lumière se déplace sur des trajectoires traversant des milieux d'indices réfraction différents, les différents temps de parcours ne sont pas égaux aux longueurs correspondantes multipliées par un nombre réel unique, indépendant des trajectoires. Cela reste vrai sur la trajectoire unique d'un rayon de lumière traversant plusieurs milieux : pour une même longueur considérée le long de la trajectoire, le temps de parcours pourra être différent selon la portion de trajectoire sur laquelle la longueur est prise. Je peux résumer cela d'une phrase :
 
-<ins>Sur l'ensemble des cas,</ins><strong> le temps de parcours n'est pas proportionnel à la distance parcourue.</strong>
+*Sur l'ensemble des cas*, **le temps de parcours n'est pas proportionnel à la distance parcourue.**
 
-Or la grandeur physique importante associée à un parcours entre deux points de l'espace, pour savoir si la lumière choisira ce parcours plutôt q'un autre, sera le temps de parcours. Cependant lorsque je visualise mentalement une trajectoire, je vois une ligne à laquelle j'associe intuitivement une longueur. <ins>Comment travailler mathématiquement avec une grandeur physique homogène à une longueur, mais qui aurait les mêmes propriétés que le temps de parcours pour décrire la propagation de la lumière </ins>? Une telle grandeur a été définie en physique, et est grandement utilisée en optique géométrique, optique ondulatoire, électromagnétisme, et elle est nommée "<strong>chemin optique</strong> noté usuellement "<strong> $\delta_o$</strong>".
+Or la grandeur physique importante associée à un parcours entre deux points de l'espace, pour savoir si la lumière choisira ce parcours plutôt q'un autre, sera le temps de parcours. Cependant lorsque je visualise mentalement une trajectoire, je vois une ligne à laquelle j'associe intuitivement une longueur. *Comment travailler mathématiquement avec une grandeur physique homogène à une longueur, mais qui aurait les mêmes propriétés que le temps de parcours pour décrire la propagation de la lumière *? Une telle grandeur a été définie en physique, et est grandement utilisée en optique géométrique, optique ondulatoire, électromagnétisme, et elle est nommée " **chemin optique** noté usuellement " **$\delta_o$**".
 
-Le chemin optique $\delta_o$ d'un parcours donné $\Gamma_o$ entre deux points A et B de l'espace est <ins>homogène à une longueur</ins>. Son <strong>unité (S.I.)</strong> (son unité dans le Système International d'unités) est donc le "<ins>mètre</ins>".
+Le chemin optique $\delta_o$ d'un parcours donné $\Gamma_o$ entre deux points A et B de l'espace est *homogène à une longueur*. Son **unité (S.I.)** (son unité dans le Système International d'unités) est donc le "*mètre*".
 
-Pour tout segment de droite pris entre deux points infiniment proche dans l'espace, son chemin optique infinitésimal (ou élémentaire) <strong>$\mathrm{d}\delta$</strong> est égal à sa <ins>longueur euclidienne $\mathrm{d}s$ multipliée par la valeur de l'indice de réfraction $n$</ins> moyennée sur le segment infinitésimal considéré :
-<strong>$\mathrm{d}\delta\;=\;n\times \mathrm{d}s$</strong>
+Pour tout segment de droite pris entre deux points infiniment proche dans l'espace, son chemin optique infinitésimal (ou élémentaire) **mathrm{d}\delta$** est égal à sa *longueur euclidienne $\mathrm{d}s$ multipliée par la valeur de l'indice de réfraction $n$* moyennée sur le segment infinitésimal considéré :
+
+**$\mathrm{d}\delta\;=\;n\times \mathrm{d}s$**
 
 Le chemin optique $\delta$ d'un parcours donné $\Gamma_o$ entre deux point de l'espace est simplement la somme des chemins optiques infinitésimaux intégrés sur ce parcours :
-<strong>$\delta = \int_{\Gamma_o}\mathrm{d}\delta= \int_{\Gamma_o}n\cdot \mathrm{d}s$</strong>
 
-Quelques soient deux points donnés A et B de l'espace, et quelque soit le parcours considéré entre ces deux points, le <strong>chemin optique</strong> sera <ins>toujours égal au temps de parcours de la lumière sur ce parcours divisé par la vitesse de la lumière dans le vide $c$ </ins>qui est une constante universelle de la nature :
-<strong>$\mathrm{d}\delta\;=\;\frac{ds}{c}$
-$\delta = \int_{S_{AB}}n\cdot\mathrm{d}s\;=\;\int_{S_{AB}}\frac{c}{v}\cdot\mathrm{d}s$
-$\hspace{1cm}= c\;\int_{S_{AB}}\frac{\mathrm{d}s}{v} =\;c\;\tau$</strong>
+**$\delta = \int_{\Gamma_o}\mathrm{d}\delta= \int_{\Gamma_o}n\cdot \mathrm{d}s$**
 
+Quelques soient deux points donnés A et B de l'espace, et quelque soit le parcours considéré entre ces deux points, le **chemin optique** sera *toujours égal au temps de parcours de la lumière sur ce parcours divisé par la vitesse de la lumière dans le vide $c$ *qui est une constante universelle de la nature :
+
+**$\mathrm{d}\delta\;=\;\frac{ds}{c}$
+$\delta = \int_{S_{AB}}n\cdot\mathrm{d}s\;=\;\int_{S_{AB}}\frac{c}{v}\cdot\mathrm{d}s$
+$\hspace{1cm}= c\;\int_{S_{AB}}\frac{\mathrm{d}s}{v} =\;c\;\tau$**
 
-<br>
-<ul class="exemple">Je peux maintenant considérer un rayon lumineux se propageant d'un point A à un point B, et lui imposer au cours de sa trajectoire entre A et B d'interagir avec un système optique. Je peux ensuite considérer l'ensemble des chemins possibles (ils sont en nombre infini en optique) entre A et B, et considérer une application f qui à chaque chemin de cet ensemble associe une grandeur physique particulière. En optique géométrique, les deux grandeurs physiques intéressantes sont le temps de parcours et le chemin optique. La question est :<br>
-Le trajet réellement suivi par la lumière dans chaque cas correspond-t-il à un chemin défini par un point particulier de la fonction f ?. Les points particuliers qui vont m'intéresser en optique géométrique sont appelés en mathématiques les points stationnaires.</ul>
+!!! Je peux maintenant considérer un rayon lumineux se propageant d'un point A à un point B, et lui imposer au cours de sa trajectoire entre A et B d'interagir avec un système optique. Je peux ensuite considérer l'ensemble des chemins possibles (ils sont en nombre infini en optique) entre A et B, et considérer une application f qui à chaque chemin de cet ensemble associe une grandeur physique particulière. En optique géométrique, les deux grandeurs physiques intéressantes sont le temps de parcours et le chemin optique. La question est :
+!!!
+!!! Le trajet réellement suivi par la lumière dans chaque cas correspond-t-il à un chemin défini par un point particulier de la fonction f ?
+!!!
+!!! Les points particuliers qui vont m'intéresser en optique géométrique sont appelés en mathématiques les points stationnaires.
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+++ b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/02.geometrical-optics-foundings/04.fermat-principle/02.fermat-principle-overview/cheatsheet.en.md
@@ -3,39 +3,94 @@ title: 'The Fermat''s principle F'
 media_order: stationnarite3_650.jpg
 ---
 
-#####Chemin optique
+#### Optical path
 
-<strong>chemin optique</strong><ins> $\delta$</ins>&nbsp;&nbsp;&nbsp;=&nbsp;&nbsp;&nbsp;
-<strong>longueur euclidienne</strong><ins> $s$ </ins>&nbsp;&nbsp;X&nbsp;&nbsp;&nbsp;<strong>indice de réfraction</strong><ins> $n$</ins>
+**optical path** *$\delta$* &nbsp;&nbsp;&nbsp;&nbsp; $=$
+**euclidean length** *$s$* &nbsp;&nbsp;&nbsp; $\times$ &nbsp;&nbsp; **refractive index** *$n$*
 
-* <strong>$\Gamma$</strong> : <ins>chemin ( = ligne continue ) entre 2 points fixes A et B</ins>
-* <strong>$\mathrm{d}s_P$</strong> : <ins>élément de longueur infinitésimal au point P sur le chemin $\Gamma$</ins>
-* <strong>$\mathrm{d}\delta_P$</strong> : <ins>chemin optique infinitésimal au point P sur le chemin $\Gamma$</ins>
+* **$\Gamma$** : *path (solid line) between 2 fixed points A and B*
+* **$\mathrm{d}s_P$** : *element of infinitesimal length at point P on path $\Gamma$*
+* **$ n_P$** : *refractive index at point P*
+* **$\mathrm{d}\delta_P$** : *infinitesimal optical path at point P on path $\Gamma$*
 
-Chemin optique le long d'un chemin entre 2 points fixes A et B :
+Optical path along a path between 2 fixed points A and B :
+**$\delta\;=\;\int_{P \in \Gamma}\mathrm{d}\delta_P\;=\;\int_{P \in \Gamma}n_P\cdot\mathrm{d}s_P$**
 
-<strong>$\delta\;=\;\int_{P \in \Gamma}\mathrm{d}\delta_P\;=\;\int_{P \in \Gamma}n_P\cdot\mathrm{d}s_P$</strong>
+* **$\delta$** $=\int_{\Gamma}n\cdot\mathrm{d}s\;=\;\int_{\Gamma}\frac{c}{v}\cdot\mathrm{d}s$ = $c\;\int_{\Gamma}\frac{\mathrm{d}s}{v}$ = *$\;c\;\tau$*
+* **$\delta$** is *proportional to the travel time*.
 
-* <strong>$\delta$</strong> $\;=\int_{\Gamma}n\cdot\mathrm{d}s\;=\;\int_{\Gamma}\frac{c}{v}\cdot\mathrm{d}s$ = $c\;\int_{\Gamma}\frac{\mathrm{d}s}{v}$ = <ins>$\;c\;\tau$</ins>
-* <strong>$\delta$</strong> est <ins>proportionnel au temps de parcours</ins>.
 
-toto
+#### Stationarity
 
-#####Stationnarité d'un chemin
-
-* <strong>$\Gamma_o$</strong> : <ins>chemin entre 2 points fixes A et B</ins>
-* <strong>$\lambda_i$ </strong> : <ins>paramètres définissant un chemin</ins>
-* <strong>${\Large\tau}$ </strong> : <ins>grandeur physique caractérisant un chemin</ins>
+* **$\Gamma_o$** : *path between two fixed point A and B*
+* **$\lambda_i$** : *parameter that defines a path*
+* **${\Large\tau}$** : *grandeur physique caractérisant un chemin*
  
-<strong>${\Large\tau}(\Gamma_o)$ stationnaire &nbsp;&nbsp;
-${\Longleftrightarrow}\:\:\:\:\:\mathrm{d}{\Large\tau}(\Gamma_o)=\sum_i\frac{\partial{\large\tau}}{\partial\lambda_i}(\Gamma_o)\;\mathrm{d}\lambda_i=0$</strong>
+**${\Large\tau}(\Gamma_o)$ stationnaire &nbsp;&nbsp;
+${\Longleftrightarrow}\:\:\:\:\:\mathrm{d}{\Large\tau}(\Gamma_o)=\sum_i\frac{\partial{\large\tau}}{\partial\lambda_i}(\Gamma_o)\;\mathrm{d}\lambda_i=0$**
+
+![](stationnarite3_400.jpg)
+
+#### Fermat's principle
+
+**Between two points** of its path, a **ray of light** follows **the path(s)** that present(s) a *stationary travelling time*.
+
+or (equivalent )
+
+**Between two points** of its path, a **ray of light** follows **the path(s)** that present(s) a *stationary optical path*.
+
+
+#### Examples 
+
+##### Spherical concave mirror
+
+* **A** : *point source* that emits or diffuses lights in all directions.
+* **B** : *fixed point in space*.
+
+For this mirror, **according to the positions of points A and B** :
+
+* **several extrema** : here *2 maxima* et *1 minimum* **$\Longrightarrow$ several ligh rays**from A go through B : here *3 rays* :
+
+<!--à finaliser si possible :
+1) DIFFICILE : mise au point automatique : largeur 100% de l'écran et bonne hauteur 'difficile cela' de toute application geogebra.
+2) TRES DIFFICILE : faire en sorte que en mode connexion, l'image animée correspondante au geogebra s'affiche tant que l'animation iframe geogebra n'est pas téléchargée, puis elle ne s'affiche pas lorsque le geogeba est prêt, et que ce soit elle qui s'affiche en mode "hors connexion".-->
+
+[Click here for geogebra animation](https://www.geogebra.org/material/iframe/id/syegm6gp)
+
+<!--image gif correspondante-->
+![](Fermat_mir_3ray_650.gif)
+
+!!!! *Be careful* :<br>
+!!!! Understand this example of application of Fermat's theorem. It says that *the 3 trajectories drawn between A and B satisfy the Fermat's principle* and therefore *are possible trajectories between these two points* to the exclusion of any other trajectory. If point A is a point source that emits light in all directions, then these 3 paths will be followed by light. If a ray goes through A with one of these 3 directions, then the corresponding trajectory will be realized.
+!!!! *But* the points *A and B are not conjugate points within th meaning of paraxial optics* : B is not the image point of point source A, and vice versa.
+!!! It is the same for all the animations of this chapter "examples".
+
+* other positions of A and B : **1 minimum $\Longrightarrow$ 1 unique ray** from A goes trhough B .
+
+![](Fermat_mir_1ray_min_650.gif)
+
+* other positions of A and B : **1 maximum $\Longrightarrow$ 1 unique ray** from A goes trhough B .
+
+![](Fermat_mir_1ray_max_650.gif)
+
+##### Elliptical concave mirror. 
+
+* Elliptical mirror : mirror whose surface is part of an ellipsoid of revolution.
+
+! <details markdown=1>
+! <summary>
+! Ellipsoid and ellipsoid of revolution
+! </summary>
+! Est-il nécessaire de rapeller ici ce que sont les ellipsoïde et ellipsoïde de révolution? rappel en texte? ou 2 liens vers Wikipédia? ou lien vers une autre page m3p2 sur les quadriques en géométrie euclidienne (page encore à créer) ? Si oui, dans une partie Beyond, parler du miroir elliptique concave acoustique, c'est impressionnant quand on le vit.
+! </details>
 
-![](stationnarite3_650.jpg)
+* **between the two geometrical foci"** F et F' of an elliptical mirror, **all path are stationary** : they have the same optical path<br>
+**$\Longrightarrow$** : *all the rays coming from one of the two foci and intercepting the mirror converge towards the second geometric focus *.
 
-#####Principe de Fermat
+[Click here for geogebra animation](https://www.geogebra.org/m/aaus5dpr)
 
-<strong>Entre 2 points</strong> de son parcours, un <strong>rayon de lumière</strong> suit <strong>"le" ou "les chemins"</strong> qui présentent un <ins>temps de parcours stationnaire</ins>.
+![](fermat_mir_elliptique_650.gif)
 
-ou ( équivalent )
+!!!! *BE CAREFUL* : 
+!!! the "geometric foci" of the ellipsoid of revolution, the "geometric surface" in which the surface of the elliptical mirror is inscribed, do not correspond to the optical foci (the two focal points)  of the elliptical mirror as they will be defined in the "optical sense" of the word "focus" in the remainder of this course. 
 
-<strong>Entre 2 points</strong>  de son parcours, la <strong>lumière</strong>  suit <strong>"le" ou "les chemins"</strong>  qui présentent un <ins>chemin optique stationnaire</ins>.
diff --git a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/02.geometrical-optics-foundings/04.fermat-principle/02.fermat-principle-overview/cheatsheet.es.md b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/02.geometrical-optics-foundings/04.fermat-principle/02.fermat-principle-overview/cheatsheet.es.md
index 88c343e0b..e9c9f2a97 100644
--- a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/02.geometrical-optics-foundings/04.fermat-principle/02.fermat-principle-overview/cheatsheet.es.md
+++ b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/02.geometrical-optics-foundings/04.fermat-principle/02.fermat-principle-overview/cheatsheet.es.md
@@ -3,37 +3,102 @@ title: 'El principio de Fermat F'
 media_order: modes-1-840-tr.png
 ---
 
-#####Chemin optique
+A  TRADUCIR
 
-<strong>chemin optique</strong><ins> $\delta$</ins>&nbsp;&nbsp;&nbsp;=&nbsp;&nbsp;&nbsp;
-<strong>longueur euclidienne</strong><ins> $s$ </ins>&nbsp;&nbsp;X&nbsp;&nbsp;&nbsp;<strong>indice de réfraction</strong><ins> $n$</ins>
+#### Chemin optique
 
-* <strong>$\Gamma$</strong> : <ins>chemin ( = ligne continue ) entre 2 points fixes A et B</ins>
-* <strong>$\mathrm{d}s_P$</strong> : <ins>élément de longueur infinitésimal au point P sur le chemin $\Gamma$</ins>
-* <strong>$\mathrm{d}\delta_P$</strong> : <ins>chemin optique infinitésimal au point P sur le chemin $\Gamma$</ins>
+**chemin optique**   *$\delta$* &nbsp;&nbsp;&nbsp;=&nbsp;&nbsp;&nbsp;
+**longueur euclidienne** *$s$* &nbsp;&nbsp;X&nbsp;&nbsp;&nbsp;**indice de réfraction** *$n$* 
+
+* **$\Gamma$** :  *chemin* ( = ligne continue ) *entre 2 points fixes* A et B 
+* **$\mathrm{d}s_P$** :  *élément de longueur infinitésimal* au point P sur le chemin $\Gamma$
+* **$n_P$** :  *indice de réfraction* au point P
+* **$\mathrm{d}\delta_P$** :  *chemin optique infinitésimal* au point P sur le chemin $\Gamma$
 
 Chemin optique le long d'un chemin entre 2 points fixes A et B :
 
-<strong>$\delta\;=\;\int_{P \in \Gamma}\mathrm{d}\delta_P\;=\;\int_{P \in \Gamma}n_P\cdot\mathrm{d}s_P$</strong>
+*$\delta\;=\;\int_{P \in \Gamma}\mathrm{d}\delta_P\;=\;\int_{P \in \Gamma}n_P\cdot\mathrm{d}s_P$*
+
+* **$\delta$** $\;=\int_{\Gamma}n\cdot\mathrm{d}s\;=\;\int_{\Gamma}\frac{c}{v}\cdot\mathrm{d}s$ = $c\;\int_{\Gamma}\frac{\mathrm{d}s}{v}$ =  *$\;c\;\tau$* 
+* **$\delta$** est  *proportionnel au temps de parcours* .
 
-* <strong>$\delta$</strong> $\;=\int_{\Gamma}n\cdot\mathrm{d}s\;=\;\int_{\Gamma}\frac{c}{v}\cdot\mathrm{d}s$ = $c\;\int_{\Gamma}\frac{\mathrm{d}s}{v}$ = <ins>$\;c\;\tau$</ins>
-* <strong>$\delta$</strong> est <ins>proportionnel au temps de parcours</ins>.
 
-#####Stationnarité d'un chemin
+#### Stationnarité d'un chemin
 
-* <strong>$'\Gamma_o'$</strong> : <ins>chemin entre 2 points fixes A et B</ins>
-* <strong>$\lambda_i$ </strong> : <ins>paramètres définissant un chemin</ins>
-* <strong>${\Large\tau}$ </strong> : <ins>grandeur physique caractérisant un chemin</ins>
+* **$\Gamma_o$** :  *chemin entre 2 points fixes* A et B
+* **$\lambda_i$ ** :  *paramètres définissant un chemin* 
+* **${\Large\tau}$ ** :  *grandeur physique caractérisant un chemin* 
  
-<strong>${\Large\tau}(\Gamma_o)$ stationnaire &nbsp;&nbsp;
-${\Longleftrightarrow}\:\:\:\:\:\mathrm{d}{\Large\tau}(\Gamma_o)=\sum_i\frac{\partial{\large\tau}}{\partial\lambda_i}(\Gamma_o)\;\mathrm{d}\lambda_i=0$</strong>
+**${\Large\tau}(\Gamma_o)$ stationnaire &nbsp;&nbsp;
+${\Longleftrightarrow}\:\:\:\:\:\mathrm{d}{\Large\tau}(\Gamma_o)=\sum_i\frac{\partial{\large\tau}}{\partial\lambda_i}(\Gamma_o)\;\mathrm{d}\lambda_i=0$**
 
-![](stationnarite3_650.jpg)
+![](stationnarite3_400.jpg)
 
-#####Principe de Fermat
+#### Principe de Fermat
 
-<strong>Entre 2 points</strong> de son parcours, un <strong>rayon de lumière</strong> suit <strong>"le" ou "les chemins"</strong> qui présentent un <ins>temps de parcours stationnaire</ins>.
+**Entre 2 points** de son parcours, un **rayon de lumière** suit **"le" ou "les chemins"** qui présentent un  *temps de parcours stationnaire* .
 
 ou ( équivalent )
 
-<strong>Entre 2 points</strong>  de son parcours, la <strong>lumière</strong>  suit <strong>"le" ou "les chemins"</strong>  qui présentent un <ins>chemin optique stationnaire</ins>.
+**Entre 2 points**  de son parcours, la **lumière**  suit **"le" ou "les chemins"**  qui présentent un  *chemin optique stationnaire* .
+
+#### Exemples 
+
+##### Miroir sphérique concave
+
+* **A** : *source ponctuelle* émettant lumière dans toutes les directions.
+* **B** : *point fixe de l'espace*.
+
+Pour ce miroir, **selon les positions des points A et B** :
+
+* **Plusieurs extrema** : ici *2 maxima* et *1 minimum* **$\Longrightarrow$ plusieurs rayons** issus de A passent par B : ici *3 rayons* :
+
+<!--à finaliser si possible :
+1) DIFFICILE : mise au point automatique : largeur 100% de l'écran et bonne hauteur 'difficile cela' de toute application geogebra.
+2) TRES DIFFICILE : faire en sorte que en mode connexion, l'image animée correspondante au geogebra s'affiche tant que l'animation iframe geogebra n'est pas téléchargée, puis elle ne s'affiche pas lorsque le geogeba est prêt, et que ce soit elle qui s'affiche en mode "hors connexion".-->
+
+[Cliquer pour animation geogebra](https://www.geogebra.org/material/iframe/id/syegm6gp)
+
+![](Fermat_mir_3ray_650.gif)
+
+!!!! *ATTENTION* :
+!!!! Comprenez bien cet exemple d'application du théorème de Fermat. Il dit que *les 3 trajectoires tracées entre A et B vérifient le principe de Fermat* et donc *sont des trajectoires possibles entre ces deux points* à l'exclusion de toute autre trajectoire. Si le point A est une source ponctuelle qui émet de la lumière dans toutes les directions, alors ces 3 trajectoires seront parcourues par la lumière. Si un rayon passe par A avec l'une de ces 3 inclinaisons, alors la trajectoire correspondante sera réalisée.
+!!!! *Mais* les points *A et B ne sont pas des points conjugués au sens de l'optique paraxiale* : A n'est pas l'image de B, et vice versa.
+!!! Il en est de même pour toutes les animations de ce chapitre "exemples".
+
+* autres positions de A et B : **1 minimum $\Longrightarrow$ 1 rayon unique** issu de A passe par B .
+
+![](Fermat_mir_1ray_min_650.gif)
+
+* autres positions de A et B : **1 maximum $\Longrightarrow$ 1 rayon unique** issu de A passe par B .
+
+![](Fermat_mir_1ray_max_650.gif)
+
+##### Miroir elliptique concave
+
+* Miroir elliptique : miroir dont la surface s'inscrit dans un ellipsoïde de révolution.
+
+! <details markdown=1>
+! <summary>
+! Ellipsoïde et ellipsoïde de révolution
+! </summary>
+! Est-il nécessaire de rapeller ici ce que sont les ellipsoïde et ellipsoïde de révolution? rappel en texte? ou 2 liens vers Wikipédia? ou lien vers une autre page m3p2 sur les quadriques en géométrie euclidienne (page encore à créer) ? Si oui, dans une partie Beyond, parler du miroir elliptique concave acoustique, c'est impressionnant quand on le vit.
+! </details>
+
+* **entre les deux "foyers géométriques"** F et F' d'un miroir elliptique, **tous les chemins interceptant le miroir sont stationnaires** : ils ont le même chemin optique<br>
+**$\Longrightarrow$** : *tous les rayons issus de l'un des foyers géométriques et interceptant le miroir convergent vers le second foyer géométrique*.
+
+[Click here for geogebra animation](https://www.geogebra.org/m/aaus5dpr)
+
+![](fermat_mir_elliptique_650.gif)
+
+!!!! *ATTENTION* : 
+!!! les "foyers géométriques" de l'ellipsoïde de révolution, "surface géométrique" dans laquelle s'inscrit la surface du miroir elliptique, ne correspondent pas aux "foyers" du miroir elliptique tels qu'ils seront définis au "sens optique" du terme dans la suite de ce cours.
+
+
+
+
+
+
+
+
diff --git a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/02.geometrical-optics-foundings/04.fermat-principle/02.fermat-principle-overview/cheatsheet.fr.md b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/02.geometrical-optics-foundings/04.fermat-principle/02.fermat-principle-overview/cheatsheet.fr.md
index 8167739df..21ada4e5e 100644
--- a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/02.geometrical-optics-foundings/04.fermat-principle/02.fermat-principle-overview/cheatsheet.fr.md
+++ b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/02.geometrical-optics-foundings/04.fermat-principle/02.fermat-principle-overview/cheatsheet.fr.md
@@ -27,8 +27,8 @@ Chemin optique le long d'un chemin entre 2 points fixes A et B :
 * **$\lambda_i$ ** :  *paramètres définissant un chemin* 
 * **${\Large\tau}$ ** :  *grandeur physique caractérisant un chemin* 
  
-*${\Large\tau}(\Gamma_o)$ stationnaire &nbsp;&nbsp;
-${\Longleftrightarrow}\:\:\:\:\:\mathrm{d}{\Large\tau}(\Gamma_o)=\sum_i\frac{\partial{\large\tau}}{\partial\lambda_i}(\Gamma_o)\;\mathrm{d}\lambda_i=0$*
+**${\Large\tau}(\Gamma_o)$ stationnaire &nbsp;&nbsp;
+${\Longleftrightarrow}\:\:\:\:\:\mathrm{d}{\Large\tau}(\Gamma_o)=\sum_i\frac{\partial{\large\tau}}{\partial\lambda_i}(\Gamma_o)\;\mathrm{d}\lambda_i=0$**
 
 ![](stationnarite3_400.jpg)
 
@@ -55,15 +55,12 @@ Pour ce miroir, **selon les positions des points A et B** :
 1) DIFFICILE : mise au point automatique : largeur 100% de l'écran et bonne hauteur 'difficile cela' de toute application geogebra.
 2) TRES DIFFICILE : faire en sorte que en mode connexion, l'image animée correspondante au geogebra s'affiche tant que l'animation iframe geogebra n'est pas téléchargée, puis elle ne s'affiche pas lorsque le geogeba est prêt, et que ce soit elle qui s'affiche en mode "hors connexion".-->
 
-<iframe width="auto" height="auto" sandbox="allow-same-origin allow-scripts" src="https://www.geogebra.org/material/iframe/id/syegm6gp" frameborder="0" allowfullscreen></iframe>
+[Cliquer pour animation geogebra](https://www.geogebra.org/material/iframe/id/syegm6gp)
 
-_smartphone : Passer en mode "paysage" pour voir les animations geoegbra_
-<iframe width="640" height="360" sandbox="allow-same-origin allow-scripts" src="https://www.geogebra.org/material/iframe/id/syegm6gp" frameborder="0" allowfullscreen></iframe>
-<!--image gif correspondante-->
 ![](Fermat_mir_3ray_650.gif)
 
 !!!! *ATTENTION* :
-!!!! Comprenez bien cette exemple d'application du théorème de Fermat. Il dit que *les 3 trajectoires tracées entre A et B vérifient le principe de Fermat* et donc *sont des trajectoires possibles entre ces deux points* à l'exclusion de toute autre trajectoire. Si le point A est une source ponctuelle qui émet de la lumière dans toutes les directions, alors ces 3 trajectoires seront parcourues par la lumière. Si un rayon passe par A avec l'une de ces 3 inclinaisons, alors la trajectoire correspondante sera réalisée.
+!!!! Comprenez bien cet exemple d'application du théorème de Fermat. Il dit que *les 3 trajectoires tracées entre A et B vérifient le principe de Fermat* et donc *sont des trajectoires possibles entre ces deux points* à l'exclusion de toute autre trajectoire. Si le point A est une source ponctuelle qui émet de la lumière dans toutes les directions, alors ces 3 trajectoires seront parcourues par la lumière. Si un rayon passe par A avec l'une de ces 3 inclinaisons, alors la trajectoire correspondante sera réalisée.
 !!!! *Mais* les points *A et B ne sont pas des points conjugués au sens de l'optique paraxiale* : A n'est pas l'image de B, et vice versa.
 !!! Il en est de même pour toutes les animations de ce chapitre "exemples".
 
@@ -89,6 +86,8 @@ _smartphone : Passer en mode "paysage" pour voir les animations geoegbra_
 * **entre les deux "foyers géométriques"** F et F' d'un miroir elliptique, **tous les chemins interceptant le miroir sont stationnaires** : ils ont le même chemin optique<br>
 **$\Longrightarrow$** : *tous les rayons issus de l'un des foyers géométriques et interceptant le miroir convergent vers le second foyer géométrique*.
 
+[Cliquez ici pour animation geogebra](https://www.geogebra.org/m/aaus5dpr)
+
 ![](fermat_mir_elliptique_650.gif)
 
 !!!! *ATTENTION* : 
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@@ -1,5 +1,39 @@
 ---
-title: 'new course  : overview'
+title: 'The 5 laws of geometrical optics'
+media_order: 'interaction_lumiere_surface_3.gif,Opt_Geo_refle_lim.gif,Loi_reflexion.png,Opt_Geo_loi_refrac.jpg'
 ---
 
-new course  : overview
\ No newline at end of file
+###The 5 laws of light rays optics
+
+**Fermat's principle $\Longrightarrow$ 5 las of light rays optics**
+
+
+##### Propagation of light is rectilinear
+
+In an homogenous and isotropice medium for its optical properties, light rays propagate in straight lines.
+
+##### Light rays do not interact with each other
+
+
+##### Law of reflection
+
+<!--Comment for a fursther step :
+Rays are geometrical constructions that indicate the direction of flow of the luminous energy
+- in an homogenous medium, rays are stright lines.
+- in an isotropic medium, rays
+-->
+
+* The **reflected ray** *belongs to the incidente plane*, and is*located in the opposite side to that of the incident ray*, with regard to the normal of the surface at the impact point. **The angle of reflection $r$ is equal to the angle of incidence $i_1$ :
+
+$r=i_1$**
+
+![](Loi_reflexion.png)
+
+##### Law of refraction
+
+![](Opt_Geo_loi_refrac.jpg)
+
+[](https://www.geogebra.org/material/iframe/id/bg5ewxee)
+
+
+
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@@ -1,5 +1,320 @@
 ---
-title: 'new course : main'
+title: 'Towards thin lenses'
 ---
 
-new course : main
\ No newline at end of file
+<!--### Lenses classification
+
+A simple lens is a volume made with a transparent substance of refractive index $n$, that present a symetry of revolution around an axis called the optical axis of the lens. This volume is bound by two polished surfaces that can be both curved, or one curved and the other plane. The curved surfaces which have to 
+
+The line joining the centres of the spheres making up the lens surfaces is called the axis of the lens.-->
+
+### Towards thin lenses
+
+! *As part of M3P2*, (UNAL-Manizales?, UdG?,) *INSA-Toulouse*, the results of this chapter "Thick lens" are not to be memorized or known. On the other hand, *understanding the reasoning* is important.
+
+This chapter is a **necessary step** at level foothills, for me *to deeply understand and better master thin lenses*. It will allow me to :
+* understand that the thin lens equation and the expression of transverse magnification for the thin lens result from a *second level of approximation* after the paraxial approximation.
+* know *in what situations* are these two equations *correct and can be used*, or *wrong and must be modified*. For example are they still correct when :<br>
+\- the thin lens is surrounded by a medium of spectral index value different than 1?<br>
+\- both media in each side of the lens have different spectral index values?
+* have a good introduction to the next main chapter *centered optical systems* because it is my first description of such a system. In particular I will understand the *necesity and requirement for the new concepts* of "principal planes and points, and nodal points" in the next chapter to characterize simply such optical centered system and calculate easily the images they realized. 
+
+##### Physical description of a "thick lens".
+
+A **thick lens** is a *centered physical system* consisting of *two spherical refracting surface that separate the lens glass of refractive index $n$* from the external mdium in each side.
+
+I study the **general case**  where the *media on both sides of the lens have different values of spectral index*.
+!!! *EXAMPLE* :  The lens can be used as a *magnifying porthole of a bathyscaphe*.
+
+##### The centered optical system "thick lens" 
+
+Im must remind the plane refracting surface viewed at level "plains". Even if the 
+
+To define an optical system, I have to define a scenario : where is the objet to be imaged or viewed ? And where is the real image of the object to be registered by a matrix sensor or where is located the eye of the observator ? This gives me the direction of propagation of the light (from object to real image or eye) through the optical system. This direction of propagation is part of the description of an optical system. In the figure above the optical systems are each time two ordered spherical refracting surfaces._
+
+A **thick lens** is a *centered optical system* consisting of *two spherical refracting surface that separate the lens glass of refractive index $n$* from the external mdium in each side. 
+
+I study the **general case**  where the *media on both sides of the lens have different values of spectral index*.
+!!! *EXAMPLE* :  The lens can be used as a *magnifying porthole of a bathyscaphe*, through which *a human in the air observes a fish in the water*.
+ 
+Le premier dioptre $DS_1$ traversé par la lumière a pour sommet $S_1$ et pour centre de courbure $C_1$, et sépare le milieu (_où se propage la lumière incidente_) d'indice de réfraction $n1$ du milieu situé entre les deux dioptres d'indice de réfraction $n$. Le deuxième dioptre $DS_2$ traversé par la lumière à la suite du premier sépare donc le milieu intermédiaire d'indice de réfraction $n$ du milieu final d'indice de réfraction $n_2$, et a pour sommet $S_2$ et pour centre de courbure $C_2$. L'espacement entre les deux dioptres est caractérisé par la distance  algébrique $\overline{S_1S_2}$. L'orientation de l'axe optique étant choisie positive selon le sens de propagation de la lumière, la distance algébrique $\overline{S_1S_2}$ est positive ($\overline{S_1S_2}>0$.
+
+Le **système centré $SO$** que constitue la lentille épaisse dans son environnement (_ses deux milieux de part et d'autre_) et ses conditions d'utilisation (_le sens considéré de propagation de la lumière à travers la lentille_) est donc **caractérisé par** :
+* l'*ordre de traversée* de ces deux dioptres par la lumière, de $DS_1$ vers $DS_2$.
+*  les *trois indices de réfraction $n_1$, $n$ et $n_2$* caractérisant respectivement le milieu de propagation de la lumière incidente sur le premier dioptre du système, le milieu intermédiaire commun aux deux dioptres et le milieu de propagation de la lumière transmise par le système.
+*  les *rayons algébriques $\overline{S_1C_1}$ et $\overline{S_2C_2}$* des deux dioptres sphériques $DS_1$ et $DS_2$.
+*  la *distance $\overline{S_1S_2}$* qui spécifie l'espacement entre les deux dioptres. 
+
+! *IMPORTANT* : si la *lentille* est *plan-convexe ou plan concave*, il suffira de reprendre les diverses expressions mathématiques trouvées et *faire tendre le rayon de courbure concerné (_faire attention au sens de propagation de la lumière_) vers l'infini* ($\overline{SC}\rightarrow\infty$). Nous retrouverions (_certes d'une façon bien compliquée_) les résultats pour une paroi transparente regardée sous incidence normale et dans les conditions de Gauss, en faisant tendre les rayons de deux dioptres vers l'infini ($\overline{S_1C_1}\rightarrow\infty$ et $\overline{S_2C_2}\rightarrow\infty$).
+
+Dans le cadre de l'optique paraxiale (_optique Gaussienne_), ce système optique est quasi-stigmatique et il donne les rayons de lumière issus du point objet $B$ un point image unique $B'$. La **position du point objet $B$ par rapport au système optique** est *déterminée par* :
+* la *distance algébrique* **$\overline{AS_1}$** entre la projection $A$ du point objet $B$ sur l'axe optique et le sommet $S_1$ du premier dioptre $DS_1$.
+* l'*élévation algébrique* **$\overline{AB}$** du point B par rapport à l'axe optique (_en choisissant préalablement un sens positif d'orientation commun à toute droite perpendiculaire à l'axe optique_).
+
+##### Calcul de l'image finale d'un objet initial de position connue
+
+La position du point $B$ est connue, grâce aux valeurs numériques de $\overline{AS_1}$ et $\overline{AB}$.
+
+!!! *EXEMPLE* : je reprends l'exemple du scientifique qui observe un poisson des abymes à travers le hublot grossissant d'un bathyscaphe. Cela donne :
+!!! * milieux extrêmes : $n_1$=$n_{eau}=4/3$ et $n_2$=$n_{air}=1$
+!!! * hublot lenticulaire : $n$=$n_{verre}=3/2$, $\overline{S_1C_1}=1 m$, $\overline{S_2C_2}= -1 m$ et $\overline{S_1S_2}=5 cm$,
+
+Pour calculer la position de l'image finale $B'$, je décompose l'action du système optique en considérant les actions successives des deux dioptres qui le constituent :
+
+Le premier dioptre $DS_1$ forme de l'objet ponctuel initial $B$ une image ponctuelle $B_{int}$. Cette image intermédiaire $B_{int}$ devient objet pour le second dioptre $DS_2$ qui en forme une image ponctuelle finale $B'$. Le point $B'$ est donc l'image ponctuelle de l'objet ponctuel $B'$ par le système optique centré $SO$ formé par les deux dioptres successifs $DS_1$ et $DS_2$.
+
+La **relation de conjugaison** genérale $`\dfrac{n_{fin}}{\overline{SA_{ima}}}-\dfrac{n_{ini}}{\overline{SA_{obj}}}=\dfrac{n_{fin}-n_{ini}}{\overline{SC}}`$ et l'**expression du grandissement transversal** générale $`\overline{M_T}=\dfrac{n_{ini}\cdot\overline{SA_{ima}}}{n_{fin}\cdot\overline{SA_{obj}}}`$ des dioptres sphériques, *appliquées au* **dioptre sphérique particulier $DS_1$** donnent :
+
+* **$`\dfrac{n}{\overline{S_1A_{int}}}-\dfrac{n_1}{\overline{S_1A}}=\dfrac{n-n_1}{\overline{S_1C_1}}`$** (équ. 1a)
+
+* **$`\overline{M_T}=\dfrac{n_1\cdot\overline{S_1A_{int}}}{n\cdot\overline{S_1A}}`$** (équ. 1b)
+
+Je peux maintenant calculer la position du point image intermédiaire $B_{int}$. De l'équation 1a je peux calculer la valeur numérique de  $\overline{S_1A_{int}}$ du point image intermédiaire $A_{int}$ (_projection orthogonale de_ $B_{int}$ _sur l'axe optique_), et de l'équation 1b la valeur numérique de l'élévation $\overline{A_{int}B_{int}}$ :
+
+$\overline{S_1A_{int}}=\dfrac{n\cdot\overline{S_1C_1}\cdot\overline{S_1A}}
+{(n-n_1)\cdot\overline{S_1A}\;+\;n_1\cdot\overline{S_1C_1}}$  (équ. 1c)
+
+par définition $\overline{M_T}=\dfrac{\overline{A_{int}B_{int}}}{\overline{AB}}$$\Longrightarrow\overline{A_{int}B_{int}}=\dfrac{n_1\cdot\overline{S_1A_{int}}}{n\cdot\overline{S_1A}}\cdot\overline{AB}$   (équ. 1d)
+
+!!! *EXEMPLE* : &nbsp;&nbsp;(_suite_)<br>
+!!! poisson :  $\overline{S_1A}= - 1 m$  et  $\overline{AB}= 7 cm$.<br>
+!!! $\Longrightarrow\overline{S_1A_{int}}=-\dfrac{9}{7} m$  et  $\overline{A_{int}B_{int}}= +8 cm$.
+
+Maintenant que la lumière à traversée le premier dioptre $DS_1$, est s'apprête à, franchir le second dioptre $DS_2$. Du point de vue du $DS_2$, les rayons incidents initiés par le point objet $B$ semblent parvenir du point intermédiaire $B_{int}$. Ce point $B_{int}$, point image pour le dioptre $DS_1$ devient point objet pour le dioptre $DS_2$.
+
+La **relation de conjugaison** et l'**expression du grandissement transversal** générales des dioptres sphériques, *appliquées au* **dioptre sphérique particulier $DS_2$** donnent :
+
+* **$`\dfrac{n_2}{\overline{S_2A'}}-\dfrac{n}{\overline{S_2A_{int}}}=\dfrac{n_2-n}{\overline{S_2C_2}}`$** (équ. 2a)
+
+* **$`\overline{M_T}=\dfrac{n\cdot\overline{S_2A'}}{n_2\cdot\overline{S_2A_{int}}}`$** (équ. 2b)
+
+Je peux calculer la valeur numérique $\overline{S_2A_{int}}$ en remarquant que
+
+$\overline{S_2A_{int}}=\overline{S_1S_2}+\overline{S_1A_{int}}=\overline{S_1A_{int}}-\overline{S_1S_2}$. 
+
+En injectant enfin $\overline{S_2A_{int}}$ et $\overline{A_{int}B_{int}}$ dans les équations 2a et 2b, je détermine les valeurs numériques $\overline{S_2A'}$ et $\overline{A'B'}$ donnant la position de l'image finale $B'$.
+
+!!! *EXEMPLE* : &nbsp;&nbsp;(_suite_)<br>
+!!! $\overline{S_2A_{int}}=-\frac{9}{7}-0.005=-1.334\:m$<br>
+!!! $\overline{S_2A'}=-1.605\:m$<br>
+!!! $\overline{A'B'}=+14.4\:cm$
+!!!
+!!! Attention, je dois donner une réponse pertinente au problème ! L'image finale n'est pas destinée à se former sur un capteur pour son enregistrement. Ce n'est pas la position de l'image par rapport au hublot ni sa taille qui sont déterminantes, mais la distance $\overline{A'O}$de l'image à l'oeil $O$ du scientifique, et l'angle apparent $\alpha$ sous lequel il voit le poisson.
+!!!
+!!! Donnée supplémentaire : l'oeil O du scientifique est situé à 10cm de la surface du hublot : $\overline{OA'}=-1.615\:m$ : donc l'image est situé devant l'oeil, le scientifique pourra la voir. De plus cette image est située plus loin que le puctum proximum, donc le scientifique pourra la voir nette. <br>
+!!! Je sais que l'image est droite, je vais travailler maintenant pour simplifier en valeurs non algébriques :<br>
+!!! $\alpha=arctg\left(\frac{A'B'}{A'O}\right)$$=arctg\left(\frac{0.144}{1.610}\right)$$=arctan(0.089)=0.089\:rad=5°$<br>
+!!! Je vois bien ici que la valeur de l'angle apparent __exprimée en radian__ est quasi identique à la valeur de sa tangente, ce qui est une condition pour considérer l'angle petit. Cela valide les conditions de Gauss considérées pour cette observation, et donc  justifie l'étude de ce problème dans le cadre de l'optique paraxiale.
+
+
+##### Calcul général de l'image finale 
+
+Si je devais chercher les deux équations qui donnent directement la position  $B'$ en fonction de la position de $B$, le calcul (qui n'est pas à faire) serait fastidieux et le résultat complexe. Il donnerait :
+
+$\overline{S_2A'}=\frac{  n_2 \cdot\overline{S_2C_2} \cdot\left(\frac{ n \cdot \overline{S_1C_1} \cdot\overline{S_1A}}{(n-n_1) \cdot\overline{S_1A}+n_1 \cdot \overline{S_1C_1}}- \overline{S_1S_2}\right)}{(n_2n)\cdot\left(\frac{n\cdot\overline{S_1C_1}\cdot\overline{S_1A}}{(n-n_1)\cdot\overline{S_1A}\;+\;n_1\cdot\overline{S_1C_1}}-\overline{S_1S_2}\right)+n\cdot \overline{S_2C_2}}$   (équ.3a)
+
+$\overline{A'B'}=\overline{AB}\times\overline{M_{T-SO}}$, avec
+
+$\overline{M_{T-SO}}=\frac{n\cdot n_1\cdot \overline{S_1C_1}}{(n-n_1)\cdot\overline{S_1A}+n_1\cdot\overline{S_1C_1}}\:\times\:$
+$\frac{\overline{S_2C_2}}{(n2-n)\cdot\left(\frac{n\cdot\overline{S_1C_1}\cdot\overline{S_1A}}{(n-n_1)\cdot\overline{S_1A}+n_1\cdot\overline{S_1C_1}}-\overline{S_1S_2}\right)-n\cdot\overline{S_2C_2}}$    (équ.3b)
+
+Ces équations sont difficiles à établir et à retenir. Essayons au moins d'établir la relation de conjugaison de type $\dfrac{n_2}{\overline{S_2A'}}-\dfrac{n_1}{\overline{S_1A}}=\cdot\cdot\cdot$
+
+##### A la recherche d'une équation de conjugaison simple pour la lentille épaisse
+
+Les équations complexes 3a et 3b sont difficiles à manipuler. le plus simples est de repartir des équations (équ.1a) et (équ.2a) où les grandeurs $\dfrac{n_1}{\overline{S_1A}}$ et $\dfrac{n_2}{\overline{S_2A'}}$ apparaissent déjà. L'addition de chaque membre des équations (équ.1a) et (équ.2a) donne :
+
+$`\dfrac{n}{\overline{S_1A_{int}}}-\dfrac{n_1}{\overline{S_1A}}+\dfrac{n_2}{\overline{S_2A'}}-\dfrac{n}{\overline{S_2A_{int}}}$$=\dfrac{n-n_1}{\overline{S_1C_1}}+\dfrac{n_2-n}{\overline{S_2C_2}}`$
+
+En ne gardant au premier membre que les termes $\dfrac{n_1}{\overline{S_1A}}$ et $\dfrac{n_2}{\overline{S_2A'}}$ j'obtiens l'équation :
+
+$\dfrac{n_2}{\overline{S_2A'}}-\dfrac{n_1}{\overline{S_1A}}$$
+\:=\:\dfrac{n-n_1}{\overline{S_1C_1}}+\dfrac{n_2-n}{\overline{S_2C_2}}+\dfrac{n}{\overline{S_2A_{int}}}-\dfrac{n}{\overline{S_1A_{int}}}$
+
+Cette équation relativement simple semble convenir, mais c'est une *équation inutile*. Certes le membre de gauche ne contient que les informations sur les conditions d'utilisation de la lentille épaisse (_les indices_ $n_1$ _et_ $n_2$ _des milieux de part et d'autre de la lentille_) et de la position de l'objet $B$ (_la distance algébrique_ $\overline{S_1A_{int}}$_, position du point_ $A$ _par rapport au sommet_ $S_1$ _du premier dioptre rencontré par la lumière_). Mais pour être utile, le membre de droite ne devrait contenir que des grandeurs caractérisant la lentille épaisse elle-même : 
+* $n-n_1$ et $n-n_2$  : indices de réfraction différentiels entre matériau constituant la lentille et les milieux extérieurs.
+* $\overline{S_1C_1}$ et $\overline{S_2C_2}$ : rayons de courbures algébriques des faces d'entrée et de sortie de la lentille. 
+* $\overline{S_1S_2}$ épaisseur de la lentille.
+Or ce terme de droite contient aussi les distances algébriques $\overline{S_1A_{int}}$ et $\overline{S_2A_{int}}$ qui concernent la position de l'image intermédiaire $A_{int}$, or cette position dépend elle-même de la position du point objet initial $A$.
+
+Lorsque les **positions des points objets et images** sont **précisées par leur distances par rapport aux sommets $S_1$ et $S_2$**, frontières physiques de la lentille épaisse avec son axe optique, il n'existe **pas d'équation simple** séparant dans un terme de gauche les conditions d'utilisation de la lentille et de positions de l'objet et de l'image, et dans un terme de droite les seules caractéristiques des deux dioptres formant la lentille épaisse. Une formule de conjugaison générale pour tout système centré sera établie au chapitre "Etude des systèmes centrés".
+
+!! *POUR  ALLER  PLUS  LOIN* :<br>
+!!
+!! Tout système centré, qu'ils soit composé de deux ou de plusieurs éléments simples centrés sur un même axe optique, pourra être caractérisé par deux plans virtuels, appelés : 
+!! * plan principal objet (P) de point d'intersection avec l'axe optique $H$
+!! * plan principal image (P') de point d'intersection avec l'axe optique $H'$
+!! qui remplaceront respectivement la face d'entrée de la première lentille ou miroir du système par la lumière, et la face de sortie de la dernière lentille ou du dernier miroir).
+!!
+!! Ces plans permettront de définir une relation de conjugaison simple de forme connue : 
+!!
+!! $\frac{n'}{\overline{H'A'}}-\frac{n}{\overline{HA}}=V$ (avec V, vergence du système dans son environnement)
+!!
+!! et serviront de référence au positionnement des points focaux objet F et image F' du système dans son environnement :
+!!
+!! $V=-\frac{n}{\overline{HF}}=\frac{n'}{\overline{H'F'}}$
+!!
+!! Contrairement à $\overline{S_1S_2}$ toujours positive qu'elle remplacera, la distance algébrique $\overline{H_1H_2}$ pourra être positive ou négative.
+
+
+### Lentille mince 
+
+Une lentille est dite **lentille mince** lorsque la *distance entre les deux sommets $S_1$ et $S_2$* de la lentille est *petite devant chacun des rayons de courbures* des deux faces.
+
+Cette condition, $\overline{S_1S_2} \ll \overline{S_1C_1}$ et $\overline{S_1S_2} \ll \overline{S_2C_2}$ me permet de faire l'approximation $\overline{S_1S_2}\rightarrow 0$ dans les diverses équations de la lentille épaisse, considérant ainsi que les sommets $S_1$ et $S_2$ se confondent en un même point O.
+
+$\overline{S_1S_2}\rightarrow 0 \:\longrightarrow\:S_1=S_2=O$$\:\longrightarrow\:\overline{S_1C_1}\rightarrow\overline{OC_1}$ et $\overline{S_2C_2}\rightarrow \overline{OC_2}$
+ 
+##### Lentille mince en milieux extrêmes différents
+
+Je peux toujours considérer la lentille mince comme un système optique composé de deux dioptres sphériques centrés, et donc reprendre l'étude initiale de la lentille épaisse, mais avec l' approximation suivante : $S_1=S_2=O$ 
+
+
+##### Lentille mince plongé dans un même milieu
+
+Je peux toujours considérer la lentille mince comme un système optique composé de deux dioptres sphériques centrés, et donc reprendre l'étude initiale de la lentille épaisse, mais avec les approximations suivantes : $S_1=S_2=O$ et $n_1=n_2=n_{ext}$
+
+
+##### Lentille mince utilisés dans l'air ou dans le vide
+
+Ce sont les conditions d'utilisation des lentilles minces dans la très grande majorité des cas. Je reprendre l'étude initiale de la lentille épaisse avec les approximations suivantes :
+
+$S_1=S_2=O$ et $n_1=n_2=1$
+
+###### Pour le **premier dioptre** :
+
+La relation de conjugaison genérale $`\dfrac{n_{fin}}{\overline{SA_{ima}}}-\dfrac{n_{ini}}{\overline{SA_{obj}}}=\dfrac{n_{fin}-n_{ini}}{\overline{SC}}`$ et l'expression du grandissement transversal générale $`\overline{M_T}=\dfrac{n_{ini}\cdot\overline{SA_{ima}}}{n_{fin}\cdot\overline{SA_{obj}}}`$ des dioptres sphériques, donnent :
+
+* **$`\dfrac{n}{\overline{OA_{int}}}-\dfrac{1}{\overline{OA}}=\dfrac{n-1}{\overline{S_1C_1}}`$** (équ. 4a)
+
+* **$`\overline{M_{T-DS1}}=\dfrac{1\cdot\overline{OA_{int}}}{n\cdot\overline{OA}}`$** (équ. 4b)
+
+###### Pour le **second dioptre** :
+
+Ces mêmes expressions générales donnent :
+
+* **$`\dfrac{1}{\overline{OA'}}-\dfrac{n}{\overline{OA_{int}}}=\dfrac{1-n}{\overline{S_2C_2}}`$** (équ. 5a)
+
+* **$`\overline{M_{T-DS2}}=\dfrac{n\cdot\overline{OA'}}{1\cdot\overline{OA_{int}}}`$** (équ. 5b)
+
+Additionner entre elles les équations 1a et 2a donne:
+
+
+$\dfrac{n}{\overline{OA_{int}}}-\dfrac{1}{\overline{OA}}+\dfrac{1}{\overline{OA'}}-\dfrac{n}{\overline{OA_{int}}}=\dfrac{n-1}{\overline{S_1C_1}}+\dfrac{1-n}{\overline{S_2C_2}}$
+
+soit 
+
+**$\dfrac{1}{\overline{OA'}}-\dfrac{1}{\overline{OA}}=(n-1)\cdot\left(\dfrac{1}{\overline{S_1C_1}}-\dfrac{1}{\overline{S_2C_2}}\right)$**  (équ. 6)
+
+**Cette équation est vraiment utile** :
+* au premier membre ne se situent que les positions des points objet et image conjugués $A$ et $A'$  sur l'axe optique relativement au point O qui positionne la lentille mince sur cet axe.
+* au second membre n'intervient que ce qui caractérise la lentille mince (_ses deux rayons algébriques de courbure_ $\overline{S_1C_1}$, $\overline{S_2C_2}$ et l'indice de réfraction $n$ du matériau qui la compose (_la position du point image intermédiaire_ $A_{int}$ _de_ $A$ _par le premier dioptre n'intervient pas_) donc ce second membre est indépendant de la position du point-objet $A$ initial. Dans ces conditions, le premier membre de cette équation définit la vergence de la lentille mince.
+
+Connaissant les caractéristiques physiques ($\overline{S_1C_1}$, $\overline{S_2C_2}$, $n$) de la lentille, cette équation xxx *peut servir à calculer la position de tout point objet $A$ ou image $A'$ connaissant la position de son point conjugué* , c'est l'**équation de conjugaison avec origine au centre O de la lentille mince** lorsque les milieux de chaque côté de la lentille ont un même indice de réfraction $n=1$. On l'appelle *relation de conjugaison de Descartes*.
+
+Les expressions des distances focales objet et image de la lentille en fonction de ses caractéristiques physiques s'obtiennent facilement.
+
+* distance focale image $\overline{OF'}$  : $\left(|\overline{OA}|\rightarrow\infty\Rightarrow A'=F'\right)$
+
+$(équ. 7)\Longrightarrow\overline{OF'}=\dfrac{\overline{S_1C_1}\cdot\overline{S_2C_2}}{(n-1)\cdot(\overline{S_2C_2}-\overline{S1_C1})}$.
+
+* distance focale objet $\overline{OF}$  : $\left(|\overline{OA'}|\rightarrow\infty\Rightarrow A=F\right)$
+
+$(équ. 8)\Longrightarrow\overline{OF}=\dfrac{\overline{S_1C_1}\cdot\overline{S_2C_2}}{(n-1)\cdot(\overline{S_1C_1}-\overline{S_2C_2})}$.
+
+Je peux réécrire la vergence (premier membre de l'équation de conjugaison) en fonction des distances focales objet et image, et je reconnais bien l'**équation de conjugaison de la lentille mince plongée un milieu d'indice de réfraction 1** apprise au niveau "collines" :
+
+**$\dfrac{1}{\overline{OA'}}-\dfrac{1}{\overline{OA}}=V=-\dfrac{1}{\overline{OF}}=\dfrac{1}{\overline{OF'}}$**  (équ. 9a)
+
+! *REMARQUE * :<br>
+! Du point de vue de l'optique paraxiale, les plans focaux objet et image sont situés à même distance de part et d'autre de la lentille mince ($\overline(OF)=-\overline(OF')$).  La lentille mince est donc optiquement symétrique. Elle est caractérisée par sa distance focale image $\overline(OF')$. Cette distance focale $\overline(OF')$ est algébrique, une distance focale négative indique une lentille mince divergente, ne distance focale positive indique une lentille mince convergente, 
+
+<!--Le principe du retour inverse de la lumière indique que si la lumière change de sens, le point focal objet devient point focal image et vice versa.-->
+
+<!-- Ce "pour aller plus lojn" suivant est long et mal calibré. Ou il est trop court, ou il est trop long. peut-être le scinder en deux, la partie courte ici, et une partie plus longue en vidéo dans la partie "Pour aller plus lojn" (partie M)-->
+!! *POUR  ALLER  PLUS  LOIN* :<br>
+!!  Du point de vue de l'optique paraxiale, toutes les lentilles minces caractérisées par une même distance focale $\overline(OF')$ sont optiquement symétrique et équivalentes, qu'elles soient physiquement symétriques ou non, et quelques soient leurs matériaux constitutifs. Ainsi les lentilles sphériques suivantes :
+!! 1. biconvexe symétrique en verre crown  (PSK) : $n=1.63$ , $\overline{S_1C_1}=+50\;cm$ , $\overline{S_2C_2}=-50\;cm$.
+!! 2. biconvexe symétrique en verre flint (BaF) : $n=1.63$ , $\overline{S_1C_1}=+50\;cm$ , $\overline{S_2C_2}=-50\;cm$. 
+!! 3. plan-convexe en verre flint (BaF) : $n=1.63$, $\overline{S_1C_1}=19.3\;cm$ côté convexe.
+!!
+!! ont une même distance focale image $\overline(OF')=+30.5\;cm$.
+!!
+!! Pourtant leurs comportements optiques réels seront légèrement différents. L'écart entre le comportement optique réel et le comportement décrit par l'optique paraxiale est dit lié aux aberrations.
+!!
+!! * Aberration chromatique : L'indice de réfraction varie légèrement avec la longueur d'onde dans le domaine visible (_selon les types de matériaux, la variation de l'indice de réfraction limité au domaine visible est modélisé par différentes fonctions de la longueur d'onde : fonctions de Cauchy, de Briot, de  Sellmeier_). Ainsi, selon la loi de Snell-Descartes, un même rayon lumineux polychromatique incident sur un dioptre avec un angle non nul donnera lieu à différents rayons (_spectre de raies_) ou un faisceau de rayons (_spectre continu_) émergents monochromatiques : c'est le phénomène de dispersion chromatique. Ainsi la position des plans focaux objet et image varient continuement sur une petite plage de distance en fonction de la longueur donde. Même dans des conditions de Gauss idéalement réalisées, un point objet diffusant une lumière blanche (de spectre continu) ne donnera pas un point image blanc, mais une petite étenddue colorée aux couleurs de l'arce-en-ciel. Ce phénomène de dispersion est bien connu dans le cas d'un prisme qui décompose la lumière incidente en un faisceau coloré, mais ce phénomène est aussi présent lorsque la lumière traverse une lentille (_même si le résultat est moins accentué grâce à sa forme_). C'est le nombre d'Abbe qui caractérise ce phénomène de dispersion chromatique : plus il est petit plus le phénomène de dispersion est important.<br>
+!! Dans l'exemple, les lentilles 1 et 2 ont mêmes rayons de courbure et un même indice de réfraction, mais le lentille en verre flint (BaF) présentera une aberration chromatique beaucoup plus importante que la lentille en verre crown (PSK). 
+!!
+!! * L'aberration géométrique : xxx.<br>
+!! Pour une lentille plan-convexe, l'aberration géométrique sera différente selon le sens de traversée de la lentille par la lumière. Les lentilles 2 et 3 sont réalisées dans un même verre et sont caractérisées par une même distance focale image, elles se comportent de façons identiques selon l'optique paraxiale. Cependant, éclairées par un même faisceau monochromatique (_pour éviter l'aberration chromatique_) sous incidence normale, c'est la lentille plan-convexe utilisée avec la lumière incidente
+
+Le grandissement transversal de la lentille mince est le produit des grandissements transversaux de chacun des deux dioptres qui composent la lentille mince. En effet :
+
+$M_T  =\dfrac{\overline{A'B'}}{\overline{AB}}=\dfrac{\overline{A_{int}B_{int}}}{\overline{AB}}\cdot\dfrac{\overline{A'B'}}{\overline{A_{int}B_{int}}}=M_{T-DS1}\cdot M_{T-DS2}$
+
+Le calcul de son expression :
+
+$M_{T-DS1}\cdot M_{T-DS2}$$=\dfrac{1\cdot\overline{OA_{int}}}{n\cdot\overline{OA}}\times\dfrac{n\cdot\overline{OA'}}{1\cdot\overline{OA_{int}}}$
+
+se simplifie en :
+
+**$M_{T-thinlens}=\dfrac{\overline{OA'}}{\overline{OA}}$**    (équ. 9b)
+
+Les équations 9a et 9b prennent le point O, centre de la lentille mince, pour référence des distances algébriques $\overline{OA}$, $\overline{OA'}$, et permettent de calculer les distances focales objet $\overline{OF}$ et image $\overline{OF'}$, et donc de postionner les points focaux $F$ et $F'$ de la lentille mince.
+
+Si c'est points sont déjà connus, alors je peux déduire une autre formule de conjugaison et une autre expression du grandissement transversal en prenant les points focaux $F$ et $F'$ pour références des distances algébriques :
+
+$\dfrac{1}{\overline{OA'}}-\dfrac{1}{\overline{OA}}=V=-\dfrac{1}{\overline{OF}}=\dfrac{1}{\overline{OF'}}$
+
+Je peux par exemple multiplier chaque membre de l'équation par  $\overline{OA}$
+
+$\dfrac{\overline{OA}}{\overline{OA'}}-1=\dfrac{\overline{OA}}{\overline{OF'}}$
+
+Puis réduire au même dénominateur le membre de gauche
+
+$\dfrac{\overline{OA}-\overline{OA'}}{\overline{OA'}}=\dfrac{\overline{OA}}{\overline{OF'}}$
+
+Je fais le produit en croix
+
+$\overline{OA'}\cdot\overline{OA}=(\overline{OA}-\overline{OA'})\cdot\overline{OF'}$
+
+Je fais apparaître $\overline{FA}$ et $\overline{F'A'}$
+
+$(\overline{OF'}+\overline{F'A'})\cdot(\overline{OF}+\overline{FA})$$=[(\overline{OF}+\overline{FA})-(\overline{OF'}+\overline{F'A'})]\cdot\overline{OF'}$
+
+Je ne garde par exemple que la distance focale image en remplaçant $\overline{OF}$ par $-\overline{OF'}$
+
+$(\overline{OF'}+\overline{F'A'})\cdot(-\overline{OF'}+\overline{FA})$$=(-\overline{OF'}+\overline{FA}-\overline{OF'}-\overline{F'A'})\cdot\overline{OF'}$
+
+J'effectue les produits
+
+$-\overline{OF'}^2+\overline{OF'}\cdot\overline{FA}-\overline{OF'}\cdot\overline{F'A'}+\overline{FA}\cdot\overline{F'A'}$$=-\overline{OF'}^2+\overline{OF'}\cdot\overline{FA}-\overline{OF'}^2-\overline{OF'}\cdot\overline{F'A'}$
+
+et je simplifie
+
+**$\overline{FA}\cdot\overline{F'A'}=-\overline{OF'}^2$**
+
+**Cette équation est vraiment utile** :
+* au premier membre ne se situent que les positions des points objet et image conjugués $A$ et $A'$  sur l'axe optique relativement aux foyers objet $F$ et image $F'$ de la lentille mince avec de chaque côté un milieu d'indice de réfraction $n=1$.
+* au second membre n'intervient que ce qui caractérise la lentille mince dans ces conditions d'utilisation (_ses distances algébriques _ $\overline{OF}$ et  $\overline{OF'}$ , positions des points focaux objet et image par rapport au point O). 
+
+Connaissant la distance focale image $\overline{OF' }$de la lentille (_attention, comme toute distance en optique géométrique, cette distance est algébrique_), cette équation xxx *peut servir à calculer la position de tout point objet $A$ ou image $A'$ connaissant la position de son point conjugué* , c'est l'**équation de conjugaison de la lentille mince, avec origines aux foyers**,
+lorsque les milieux de chaque côté de la lentille ont un même indice de réfraction $n=1$. On l'appelle *relation de conjugaison de Newton*.
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
diff --git a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/02.paraxial-optics-simple-elements/03.lens/01.lens-main/textbook.es.md b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/02.paraxial-optics-simple-elements/03.lens/01.lens-main/textbook.es.md
index f94dbfedc..b03a51ca7 100644
--- a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/02.paraxial-optics-simple-elements/03.lens/01.lens-main/textbook.es.md
+++ b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/02.paraxial-optics-simple-elements/03.lens/01.lens-main/textbook.es.md
@@ -121,7 +121,7 @@ Or ce terme de droite contient aussi les distances algébriques $\overline{S_1A_
 
 Lorsque les **positions des points objets et images** sont **précisées par leur distances par rapport aux sommets $S_1$ et $S_2$**, frontières physiques de la lentille épaisse avec son axe optique, il n'existe **pas d'équation simple** séparant dans un terme de gauche les conditions d'utilisation de la lentille et de positions de l'objet et de l'image, et dans un terme de droite les seules caractéristiques des deux dioptres formant la lentille épaisse. Une formule de conjugaison générale pour tout système centré sera établie au chapitre "Etude des systèmes centrés".
 
-!! *Pour aller plus loin*: 
+!! *POUR  ALLER  PLUS  LOIN* :<br>
 !!
 !! Tout système centré, qu'ils soit composé de deux ou de plusieurs éléments simples centrés sur un même axe optique, pourra être caractérisé par deux plans virtuels, appelés : 
 !! * plan principal objet (P) de point d'intersection avec l'axe optique $H$
@@ -145,7 +145,7 @@ Une lentille est dite **lentille mince** lorsque la *distance entre les deux som
 
 Cette condition, $\overline{S_1S_2} \ll \overline{S_1C_1}$ et $\overline{S_1S_2} \ll \overline{S_2C_2}$ me permet de faire l'approximation $\overline{S_1S_2}\rightarrow 0$ dans les diverses équations de la lentille épaisse, considérant ainsi que les sommets $S_1$ et $S_2$ se confondent en un même point O.
 
-$\overline{S_1S_2}\rightarrow 0 \:\longrightarrow\:S_1=S_2=O$$\:\longrightarrow\:$\overline{S_1C_1}\rightarrow\overline{OC_1}$ et $\overline{S_2C_2}\rightarrow \overline{OC_2}$
+$\overline{S_1S_2}\rightarrow 0 \:\longrightarrow\:S_1=S_2=O$$\:\longrightarrow\:\overline{S_1C_1}\rightarrow\overline{OC_1}$ et $\overline{S_2C_2}\rightarrow \overline{OC_2}$
  
 ##### Lentille mince en milieux extrêmes différents
 
@@ -167,17 +167,17 @@ $S_1=S_2=O$ et $n_1=n_2=1$
 
 La relation de conjugaison genérale $`\dfrac{n_{fin}}{\overline{SA_{ima}}}-\dfrac{n_{ini}}{\overline{SA_{obj}}}=\dfrac{n_{fin}-n_{ini}}{\overline{SC}}`$ et l'expression du grandissement transversal générale $`\overline{M_T}=\dfrac{n_{ini}\cdot\overline{SA_{ima}}}{n_{fin}\cdot\overline{SA_{obj}}}`$ des dioptres sphériques, donnent :
 
-* **$`\dfrac{n}{\overline{OA_{int}}}-\dfrac{1}{\overline{OA}}=\dfrac{n-1}{\overline{S_1C_1}}`$** (équ. 1a)
+* **$`\dfrac{n}{\overline{OA_{int}}}-\dfrac{1}{\overline{OA}}=\dfrac{n-1}{\overline{S_1C_1}}`$** (équ. 4a)
 
-* **$`\overline{M_T}=\dfrac{1\cdot\overline{OA_{int}}}{n\cdot\overline{OA}}`$** (équ. 1b)
+* **$`\overline{M_{T-DS1}}=\dfrac{1\cdot\overline{OA_{int}}}{n\cdot\overline{OA}}`$** (équ. 4b)
 
 ###### Pour le **second dioptre** :
 
 Ces mêmes expressions générales donnent :
 
-* **$`\dfrac{1}{\overline{OA'}}-\dfrac{n}{\overline{OA_{int}}}=\dfrac{1-n}{\overline{S_2C_2}}`$** (équ. 2a)
+* **$`\dfrac{1}{\overline{OA'}}-\dfrac{n}{\overline{OA_{int}}}=\dfrac{1-n}{\overline{S_2C_2}}`$** (équ. 5a)
 
-* **$`\overline{M_T}=\dfrac{n\cdot\overline{OA'}}{1\cdot\overline{OA_{int}}}`$** (équ. 2b)
+* **$`\overline{M_{T-DS2}}=\dfrac{n\cdot\overline{OA'}}{1\cdot\overline{OA_{int}}}`$** (équ. 5b)
 
 Additionner entre elles les équations 1a et 2a donne:
 
@@ -186,16 +186,120 @@ $\dfrac{n}{\overline{OA_{int}}}-\dfrac{1}{\overline{OA}}+\dfrac{1}{\overline{OA'
 
 soit 
 
-**$\dfrac{1}{\overline{OA'}}-\dfrac{1}{\overline{OA}}=(n-1)\cdot\left(\dfrac{1}{\overline{S_1C_1}}-\dfrac{1}{\overline{S_2C_2}}\right)$**  (équ. xxx)
+**$\dfrac{1}{\overline{OA'}}-\dfrac{1}{\overline{OA}}=(n-1)\cdot\left(\dfrac{1}{\overline{S_1C_1}}-\dfrac{1}{\overline{S_2C_2}}\right)$**  (équ. 6)
 
 **Cette équation est vraiment utile** :
 * au premier membre ne se situent que les positions des points objet et image conjugués $A$ et $A'$  sur l'axe optique relativement au point O qui positionne la lentille mince sur cet axe.
-* au second membre n'intervient que ce qui caractérise la lentille mince (_ses deux rayons algébriques de courbure_ $S_1C_1$, $S_2C_2$ et l'indice de réfraction $n$ du matériau qui la compose (_la position du point image intermédiaire_ $A_{int}$ _de_ $A$ _par le premier dioptre n'intervient pas_) donc ce second membre est indépendant de la position du point-objet $A$ initial. 
+* au second membre n'intervient que ce qui caractérise la lentille mince (_ses deux rayons algébriques de courbure_ $\overline{S_1C_1}$, $\overline{S_2C_2}$ et l'indice de réfraction $n$ du matériau qui la compose (_la position du point image intermédiaire_ $A_{int}$ _de_ $A$ _par le premier dioptre n'intervient pas_) donc ce second membre est indépendant de la position du point-objet $A$ initial. Dans ces conditions, le premier membre de cette équation définit la vergence de la lentille mince.
+
+Connaissant les caractéristiques physiques ($\overline{S_1C_1}$, $\overline{S_2C_2}$, $n$) de la lentille, cette équation xxx *peut servir à calculer la position de tout point objet $A$ ou image $A'$ connaissant la position de son point conjugué* , c'est l'**équation de conjugaison avec origine au centre O de la lentille mince** lorsque les milieux de chaque côté de la lentille ont un même indice de réfraction $n=1$. On l'appelle *relation de conjugaison de Descartes*.
+
+Les expressions des distances focales objet et image de la lentille en fonction de ses caractéristiques physiques s'obtiennent facilement.
+
+* distance focale image $\overline{OF'}$  : $\left(|\overline{OA}|\rightarrow\infty\Rightarrow A'=F'\right)$
+
+$(équ. 7)\Longrightarrow\overline{OF'}=\dfrac{\overline{S_1C_1}\cdot\overline{S_2C_2}}{(n-1)\cdot(\overline{S_2C_2}-\overline{S1_C1})}$.
+
+* distance focale objet $\overline{OF}$  : $\left(|\overline{OA'}|\rightarrow\infty\Rightarrow A=F\right)$
+
+$(équ. 8)\Longrightarrow\overline{OF}=\dfrac{\overline{S_1C_1}\cdot\overline{S_2C_2}}{(n-1)\cdot(\overline{S_1C_1}-\overline{S_2C_2})}$.
+
+Je peux réécrire la vergence (premier membre de l'équation de conjugaison) en fonction des distances focales objet et image, et je reconnais bien l'**équation de conjugaison de la lentille mince plongée un milieu d'indice de réfraction 1** apprise au niveau "collines" :
+
+**$\dfrac{1}{\overline{OA'}}-\dfrac{1}{\overline{OA}}=V=-\dfrac{1}{\overline{OF}}=\dfrac{1}{\overline{OF'}}$**  (équ. 9a)
+
+! *REMARQUE * :<br>
+! Du point de vue de l'optique paraxiale, les plans focaux objet et image sont situés à même distance de part et d'autre de la lentille mince ($\overline(OF)=-\overline(OF')$).  La lentille mince est donc optiquement symétrique. Elle est caractérisée par sa distance focale image $\overline(OF')$. Cette distance focale $\overline(OF')$ est algébrique, une distance focale négative indique une lentille mince divergente, ne distance focale positive indique une lentille mince convergente, 
+
+<!--Le principe du retour inverse de la lumière indique que si la lumière change de sens, le point focal objet devient point focal image et vice versa.-->
+
+<!-- Ce "pour aller plus lojn" suivant est long et mal calibré. Ou il est trop court, ou il est trop long. peut-être le scinder en deux, la partie courte ici, et une partie plus longue en vidéo dans la partie "Pour aller plus lojn" (partie M)-->
+!! *POUR  ALLER  PLUS  LOIN* :<br>
+!!  Du point de vue de l'optique paraxiale, toutes les lentilles minces caractérisées par une même distance focale $\overline(OF')$ sont optiquement symétrique et équivalentes, qu'elles soient physiquement symétriques ou non, et quelques soient leurs matériaux constitutifs. Ainsi les lentilles sphériques suivantes :
+!! 1. biconvexe symétrique en verre crown  (PSK) : $n=1.63$ , $\overline{S_1C_1}=+50\;cm$ , $\overline{S_2C_2}=-50\;cm$.
+!! 2. biconvexe symétrique en verre flint (BaF) : $n=1.63$ , $\overline{S_1C_1}=+50\;cm$ , $\overline{S_2C_2}=-50\;cm$. 
+!! 3. plan-convexe en verre flint (BaF) : $n=1.63$, $\overline{S_1C_1}=19.3\;cm$ côté convexe.
+!!
+!! ont une même distance focale image $\overline(OF')=+30.5\;cm$.
+!!
+!! Pourtant leurs comportements optiques réels seront légèrement différents. L'écart entre le comportement optique réel et le comportement décrit par l'optique paraxiale est dit lié aux aberrations.
+!!
+!! * Aberration chromatique : L'indice de réfraction varie légèrement avec la longueur d'onde dans le domaine visible (_selon les types de matériaux, la variation de l'indice de réfraction limité au domaine visible est modélisé par différentes fonctions de la longueur d'onde : fonctions de Cauchy, de Briot, de  Sellmeier_). Ainsi, selon la loi de Snell-Descartes, un même rayon lumineux polychromatique incident sur un dioptre avec un angle non nul donnera lieu à différents rayons (_spectre de raies_) ou un faisceau de rayons (_spectre continu_) émergents monochromatiques : c'est le phénomène de dispersion chromatique. Ainsi la position des plans focaux objet et image varient continuement sur une petite plage de distance en fonction de la longueur donde. Même dans des conditions de Gauss idéalement réalisées, un point objet diffusant une lumière blanche (de spectre continu) ne donnera pas un point image blanc, mais une petite étenddue colorée aux couleurs de l'arce-en-ciel. Ce phénomène de dispersion est bien connu dans le cas d'un prisme qui décompose la lumière incidente en un faisceau coloré, mais ce phénomène est aussi présent lorsque la lumière traverse une lentille (_même si le résultat est moins accentué grâce à sa forme_). C'est le nombre d'Abbe qui caractérise ce phénomène de dispersion chromatique : plus il est petit plus le phénomène de dispersion est important.<br>
+!! Dans l'exemple, les lentilles 1 et 2 ont mêmes rayons de courbure et un même indice de réfraction, mais le lentille en verre flint (BaF) présentera une aberration chromatique beaucoup plus importante que la lentille en verre crown (PSK). 
+!!
+!! * L'aberration géométrique : xxx.<br>
+!! Pour une lentille plan-convexe, l'aberration géométrique sera différente selon le sens de traversée de la lentille par la lumière. Les lentilles 2 et 3 sont réalisées dans un même verre et sont caractérisées par une même distance focale image, elles se comportent de façons identiques selon l'optique paraxiale. Cependant, éclairées par un même faisceau monochromatique (_pour éviter l'aberration chromatique_) sous incidence normale, c'est la lentille plan-convexe utilisée avec la lumière incidente
+
+Le grandissement transversal de la lentille mince est le produit des grandissements transversaux de chacun des deux dioptres qui composent la lentille mince. En effet :
+
+$M_T  =\dfrac{\overline{A'B'}}{\overline{AB}}=\dfrac{\overline{A_{int}B_{int}}}{\overline{AB}}\cdot\dfrac{\overline{A'B'}}{\overline{A_{int}B_{int}}}=M_{T-DS1}\cdot M_{T-DS2}$
+
+Le calcul de son expression :
+
+$M_{T-DS1}\cdot M_{T-DS2}$$=\dfrac{1\cdot\overline{OA_{int}}}{n\cdot\overline{OA}}\times\dfrac{n\cdot\overline{OA'}}{1\cdot\overline{OA_{int}}}$
+
+se simplifie en :
+
+**$M_{T-thinlens}=\dfrac{\overline{OA'}}{\overline{OA}}$**    (équ. 9b)
+
+Les équations 9a et 9b prennent le point O, centre de la lentille mince, pour référence des distances algébriques $\overline{OA}$, $\overline{OA'}$, et permettent de calculer les distances focales objet $\overline{OF}$ et image $\overline{OF'}$, et donc de postionner les points focaux $F$ et $F'$ de la lentille mince.
+
+Si c'est points sont déjà connus, alors je peux déduire une autre formule de conjugaison et une autre expression du grandissement transversal en prenant les points focaux $F$ et $F'$ pour références des distances algébriques :
+
+$\dfrac{1}{\overline{OA'}}-\dfrac{1}{\overline{OA}}=V=-\dfrac{1}{\overline{OF}}=\dfrac{1}{\overline{OF'}}$
+
+Je peux par exemple multiplier chaque membre de l'équation par  $\overline{OA}$
+
+$\dfrac{\overline{OA}}{\overline{OA'}}-1=\dfrac{\overline{OA}}{\overline{OF'}}$
+
+Puis réduire au même dénominateur le membre de gauche
+
+$\dfrac{\overline{OA}-\overline{OA'}}{\overline{OA'}}=\dfrac{\overline{OA}}{\overline{OF'}}$
+
+Je fais le produit en croix
+
+$\overline{OA'}\cdot\overline{OA}=(\overline{OA}-\overline{OA'})\cdot\overline{OF'}$
+
+Je fais apparaître $\overline{FA}$ et $\overline{F'A'}$
+
+$(\overline{OF'}+\overline{F'A'})\cdot(\overline{OF}+\overline{FA})$$=[(\overline{OF}+\overline{FA})-(\overline{OF'}+\overline{F'A'})]\cdot\overline{OF'}$
+
+Je ne garde par exemple que la distance focale image en remplaçant $\overline{OF}$ par $-\overline{OF'}$
+
+$(\overline{OF'}+\overline{F'A'})\cdot(-\overline{OF'}+\overline{FA})$$=(-\overline{OF'}+\overline{FA}-\overline{OF'}-\overline{F'A'})\cdot\overline{OF'}$
+
+J'effectue les produits
+
+$-\overline{OF'}^2+\overline{OF'}\cdot\overline{FA}-\overline{OF'}\cdot\overline{F'A'}+\overline{FA}\cdot\overline{F'A'}$$=-\overline{OF'}^2+\overline{OF'}\cdot\overline{FA}-\overline{OF'}^2-\overline{OF'}\cdot\overline{F'A'}$
+
+et je simplifie
+
+**$\overline{FA}\cdot\overline{F'A'}=-\overline{OF'}^2$**
+
+**Cette équation est vraiment utile** :
+* au premier membre ne se situent que les positions des points objet et image conjugués $A$ et $A'$  sur l'axe optique relativement aux foyers objet $F$ et image $F'$ de la lentille mince avec de chaque côté un milieu d'indice de réfraction $n=1$.
+* au second membre n'intervient que ce qui caractérise la lentille mince dans ces conditions d'utilisation (_ses distances algébriques _ $\overline{OF}$ et  $\overline{OF'}$ , positions des points focaux objet et image par rapport au point O). 
+
+Connaissant la distance focale image $\overline{OF' }$de la lentille (_attention, comme toute distance en optique géométrique, cette distance est algébrique_), cette équation xxx *peut servir à calculer la position de tout point objet $A$ ou image $A'$ connaissant la position de son point conjugué* , c'est l'**équation de conjugaison de la lentille mince, avec origines aux foyers**,
+lorsque les milieux de chaque côté de la lentille ont un même indice de réfraction $n=1$. On l'appelle *relation de conjugaison de Newton*.
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
 
-Connaissant les caractéristiques physiques ($S_1C_1$, $S_2C_2$, $n$) de la lentille, cette équation xxx *peut servir à calculer la position de tout point objet $A$ ou image $A'$ connaissant la position de son point conjugué* , c'est l'**équation de conjugaison de la lentille mince**.
 
-Cette lentille étant plongée dans un même milieu d'indice de réfraction fixé $n_{ext}=1$, la vergence comme les points focaux image et objet de la lentille peuvent être déterminés à partir des caractéristiques physiques de la lentille :
 
 
-la **relation de conjugaison de la lentille mince** vue au niveau "plaine".
 
diff --git a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/02.paraxial-optics-simple-elements/03.lens/01.lens-main/textbook.fr.md b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/02.paraxial-optics-simple-elements/03.lens/01.lens-main/textbook.fr.md
index 009db1e3e..08fc6bbc6 100644
--- a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/02.paraxial-optics-simple-elements/03.lens/01.lens-main/textbook.fr.md
+++ b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/02.paraxial-optics-simple-elements/03.lens/01.lens-main/textbook.fr.md
@@ -2,28 +2,44 @@
 title: 'nouveau cours : principal'
 ---
 
-nouveau cours : principal
-
-#### Classification des lentilles
+### Classification des lentilles
 
 A simple lens is a volume made with a transparent substance of refractive index $n$, that present a symetry of revolution around an axis called the optical axis of the lens. This volume is bound by two polished surfaces that can be both curved, or one curved and the other plane. The curved surfaces which have to 
 
-#### Lentille épaisse : 2 dioptres sphériques centrés successifs.
+### Lentille épaisse : 2 dioptres sphériques centrés successifs.
+
+! Dans la *cadre de m3p2, (UNAL-Manizales?, UdG?,) INSA-Toulouse*, les résultats de ce chapitre "Lentille épaisse" ne sont pas à mémoriser ni connaître. Par contre, *comprendre le raisonnement* est important.
 
-Considérons le système optique $SO$ constitué par deux dioptres sphériques de symétries de révolution partageant un même axe de révolution. 
+##### Le système centré "lentille épaisse" 
 
+Une **lentille épaisse** est un *système optique centré* formé de *deux dioptres sphériques séparant le verre constitutif de la lentille d'indice de réfraction $n$*. 
+
+Nous allons mener l'**étude du cas général** où les *milieux de part et d'autres de la lentille sont différents*, et pour un sens de propagation de la lumière à travers la lentille.
+!!! *EXEMPLE* :  la lentille peut être utilisé comme *hublot grossissant d'un bathyscaphe*, à travers lequel *un humain dans l'air observe un poisson dans l'eau*.
+ 
 Le premier dioptre $DS_1$ traversé par la lumière a pour sommet $S_1$ et pour centre de courbure $C_1$, et sépare le milieu (_où se propage la lumière incidente_) d'indice de réfraction $n1$ du milieu situé entre les deux dioptres d'indice de réfraction $n$. Le deuxième dioptre $DS_2$ traversé par la lumière à la suite du premier sépare donc le milieu intermédiaire d'indice de réfraction $n$ du milieu final d'indice de réfraction $n_2$, et a pour sommet $S_2$ et pour centre de courbure $C_2$. L'espacement entre les deux dioptres est caractérisé par la distance  algébrique $\overline{S_1S_2}$. L'orientation de l'axe optique étant choisie positive selon le sens de propagation de la lumière, la distance algébrique $\overline{S_1S_2}$ est positive ($\overline{S_1S_2}>0$.
 
 Le **système centré $SO$** que constitue la lentille épaisse dans son environnement (_ses deux milieux de part et d'autre_) et ses conditions d'utilisation (_le sens considéré de propagation de la lumière à travers la lentille_) est donc **caractérisé par** :
 * l'*ordre de traversée* de ces deux dioptres par la lumière, de $DS_1$ vers $DS_2$.
-*  *les trois indices de réfraction $n_1$, $n$ et $n_2$* caractérisant respectivement le milieu de propagation de la lumière incidente sur le premier dioptre du système, le milieu intermédiaire commun aux deux dioptres et le milieu de propagation de la lumière transmise par le système.
+*  les *trois indices de réfraction $n_1$, $n$ et $n_2$* caractérisant respectivement le milieu de propagation de la lumière incidente sur le premier dioptre du système, le milieu intermédiaire commun aux deux dioptres et le milieu de propagation de la lumière transmise par le système.
+*  les *rayons algébriques $\overline{S_1C_1}$ et $\overline{S_2C_2}$* des deux dioptres sphériques $DS_1$ et $DS_2$.
 *  la *distance $\overline{S_1S_2}$* qui spécifie l'espacement entre les deux dioptres. 
 
+! *IMPORTANT* : si la *lentille* est *plan-convexe ou plan concave*, il suffira de reprendre les diverses expressions mathématiques trouvées et *faire tendre le rayon de courbure concerné (_faire attention au sens de propagation de la lumière_) vers l'infini* ($\overline{SC}\rightarrow\infty$). Nous retrouverions (_certes d'une façon bien compliquée_) les résultats pour une paroi transparente regardée sous incidence normale et dans les conditions de Gauss, en faisant tendre les rayons de deux dioptres vers l'infini ($\overline{S_1C_1}\rightarrow\infty$ et $\overline{S_2C_2}\rightarrow\infty$).
+
 Dans le cadre de l'optique paraxiale (_optique Gaussienne_), ce système optique est quasi-stigmatique et il donne les rayons de lumière issus du point objet $B$ un point image unique $B'$. La **position du point objet $B$ par rapport au système optique** est *déterminée par* :
 * la *distance algébrique* **$\overline{AS_1}$** entre la projection $A$ du point objet $B$ sur l'axe optique et le sommet $S_1$ du premier dioptre $DS_1$.
 * l'*élévation algébrique* **$\overline{AB}$** du point B par rapport à l'axe optique (_en choisissant préalablement un sens positif d'orientation commun à toute droite perpendiculaire à l'axe optique_).
 
-Nous allons établir, en fonction des grandeurs caractéristiques du système étudié, la relation de conjugaison qui permet de calculer la position du point $A'$, projection orthogonale du point image $B'$ sur l'axe optique, puis l'expression du grandissement transversal pour ce point objet $B$ qui permettra de calculer l'élévation algébrique du point image $B'$. Pour cela, décomposons l'action du système optique en considérant les actions successives des deux dioptres qui le constituent :
+##### Calcul de l'image finale d'un objet initial de position connue
+
+La position du point $B$ est connue, grâce aux valeurs numériques de $\overline{AS_1}$ et $\overline{AB}$.
+
+!!! *EXEMPLE* : je reprends l'exemple du scientifique qui observe un poisson des abymes à travers le hublot grossissant d'un bathyscaphe. Cela donne :
+!!! * milieux extrêmes : $n_1$=$n_{eau}=4/3$ et $n_2$=$n_{air}=1$
+!!! * hublot lenticulaire : $n$=$n_{verre}=3/2$, $\overline{S_1C_1}=1 m$, $\overline{S_2C_2}= -1 m$ et $\overline{S_1S_2}=5 cm$,
+
+Pour calculer la position de l'image finale $B'$, je décompose l'action du système optique en considérant les actions successives des deux dioptres qui le constituent :
 
 Le premier dioptre $DS_1$ forme de l'objet ponctuel initial $B$ une image ponctuelle $B_{int}$. Cette image intermédiaire $B_{int}$ devient objet pour le second dioptre $DS_2$ qui en forme une image ponctuelle finale $B'$. Le point $B'$ est donc l'image ponctuelle de l'objet ponctuel $B'$ par le système optique centré $SO$ formé par les deux dioptres successifs $DS_1$ et $DS_2$.
 
@@ -33,7 +49,18 @@ La **relation de conjugaison** genérale $`\dfrac{n_{fin}}{\overline{SA_{ima}}}-
 
 * **$`\overline{M_T}=\dfrac{n_1\cdot\overline{S_1A_{int}}}{n\cdot\overline{S_1A}}`$** (équ. 1b)
 
-Maintenant que la lumière à traversée le premier dioptre $DS_1$, est s'apprête à, franchir le second dioptre $DS_2$. Du point de vue du $DS_2$, les rayons incidents initiés par le point objet $B$ semblent prevenir du point intermédiaire $B_{int}$. $B_{int}$ point image pour le dioptre $DS_1$ devient point objet pour le dioptre $DS_2$.
+Je peux maintenant calculer la position du point image intermédiaire $B_{int}$. De l'équation 1a je peux calculer la valeur numérique de  $\overline{S_1A_{int}}$ du point image intermédiaire $A_{int}$ (_projection orthogonale de_ $B_{int}$ _sur l'axe optique_), et de l'équation 1b la valeur numérique de l'élévation $\overline{A_{int}B_{int}}$ :
+
+$\overline{S_1A_{int}}=\dfrac{n\cdot\overline{S_1C_1}\cdot\overline{S_1A}}
+{(n-n_1)\cdot\overline{S_1A}\;+\;n_1\cdot\overline{S_1C_1}}$  (équ. 1c)
+
+par définition $\overline{M_T}=\dfrac{\overline{A_{int}B_{int}}}{\overline{AB}}$$\Longrightarrow\overline{A_{int}B_{int}}=\dfrac{n_1\cdot\overline{S_1A_{int}}}{n\cdot\overline{S_1A}}\cdot\overline{AB}$   (équ. 1d)
+
+!!! *EXEMPLE* : &nbsp;&nbsp;(_suite_)<br>
+!!! poisson :  $\overline{S_1A}= - 1 m$  et  $\overline{AB}= 7 cm$.<br>
+!!! $\Longrightarrow\overline{S_1A_{int}}=-\dfrac{9}{7} m$  et  $\overline{A_{int}B_{int}}= +8 cm$.
+
+Maintenant que la lumière à traversée le premier dioptre $DS_1$, est s'apprête à, franchir le second dioptre $DS_2$. Du point de vue du $DS_2$, les rayons incidents initiés par le point objet $B$ semblent parvenir du point intermédiaire $B_{int}$. Ce point $B_{int}$, point image pour le dioptre $DS_1$ devient point objet pour le dioptre $DS_2$.
 
 La **relation de conjugaison** et l'**expression du grandissement transversal** générales des dioptres sphériques, *appliquées au* **dioptre sphérique particulier $DS_2$** donnent :
 
@@ -41,18 +68,237 @@ La **relation de conjugaison** et l'**expression du grandissement transversal**
 
 * **$`\overline{M_T}=\dfrac{n\cdot\overline{S_2A'}}{n_2\cdot\overline{S_2A_{int}}}`$** (équ. 2b)
 
-Pour relier les positions de l'objet initial et de son image finale à travers le système optique, représentées par les distances algébriques $S_1A$ et $S_2A'$, j'additionne entre eux les membres de droite des équations (équ.1a) et (équ.2a), j'aditionne entre eux les membres de gauche de ces mêmes équations, pour obtenir l'égalité suivante :
+Je peux calculer la valeur numérique $\overline{S_2A_{int}}$ en remarquant que
+
+$\overline{S_2A_{int}}=\overline{S_1S_2}+\overline{S_1A_{int}}=\overline{S_1A_{int}}-\overline{S_1S_2}$. 
+
+En injectant enfin $\overline{S_2A_{int}}$ et $\overline{A_{int}B_{int}}$ dans les équations 2a et 2b, je détermine les valeurs numériques $\overline{S_2A'}$ et $\overline{A'B'}$ donnant la position de l'image finale $B'$.
+
+!!! *EXEMPLE* : &nbsp;&nbsp;(_suite_)<br>
+!!! $\overline{S_2A_{int}}=-\frac{9}{7}-0.005=-1.334\:m$<br>
+!!! $\overline{S_2A'}=-1.605\:m$<br>
+!!! $\overline{A'B'}=+14.4\:cm$
+!!!
+!!! Attention, je dois donner une réponse pertinente au problème ! L'image finale n'est pas destinée à se former sur un capteur pour son enregistrement. Ce n'est pas la position de l'image par rapport au hublot ni sa taille qui sont déterminantes, mais la distance $\overline{A'O}$de l'image à l'oeil $O$ du scientifique, et l'angle apparent $\alpha$ sous lequel il voit le poisson.
+!!!
+!!! Donnée supplémentaire : l'oeil O du scientifique est situé à 10cm de la surface du hublot : $\overline{OA'}=-1.615\:m$ : donc l'image est situé devant l'oeil, le scientifique pourra la voir. De plus cette image est située plus loin que le puctum proximum, donc le scientifique pourra la voir nette. <br>
+!!! Je sais que l'image est droite, je vais travailler maintenant pour simplifier en valeurs non algébriques :<br>
+!!! $\alpha=arctg\left(\frac{A'B'}{A'O}\right)$$=arctg\left(\frac{0.144}{1.610}\right)$$=arctan(0.089)=0.089\:rad=5°$<br>
+!!! Je vois bien ici que la valeur de l'angle apparent __exprimée en radian__ est quasi identique à la valeur de sa tangente, ce qui est une condition pour considérer l'angle petit. Cela valide les conditions de Gauss considérées pour cette observation, et donc  justifie l'étude de ce problème dans le cadre de l'optique paraxiale.
 
-$`\dfrac{n}{\overline{S_1A_{int}}}-\dfrac{n_1}{\overline{S_1A}}+\dfrac{n_2}{\overline{S_2A'}}-\dfrac{n}{\overline{S_2A_{int}}}$ $=\dfrac{n-n_1}{\overline{S_1C_1}}+\dfrac{n_2-n}{\overline{S_2C_2}}`$
 
-Je réorganise les élements de cette équation en gardant :
-* au premier membre les éléments caractérisant les conditions extérieures d'utilisation du système optique : $n_1$ et $n_2$ (_indices des milieux de part et d'autre de $SO$_), $\overline{S_1A}$ et $\overline{S_2A'}$ (_positions de l'objet et de l'image par rapport aux faces d'entrée et de sortie de la lentille_).
-* au second membre les éléments caractérisant le système optique lui-même : $n-n_1$ et $n-n_2$ (_indices de réfraction différentiels entre matériau constituant la lentille et les milieux extérieurs_), $\overline{S_1C_1}$ et $\overline{S_2C_2}$ (_rayons de courbures algébriques des faces d'entrée et de sortie de la lentille _), $\overline{S_2A_{int}}$ et $\overline{S_1A_{int}} $(position de l'image intermédiaire par rapports aux sommets de la lentille).
+##### Calcul général de l'image finale 
 
-et j'obtiens la **relation de conjugaison du système optique** : *lentille épaisse utilisée dans les 2 milieux extérieurs considérés* :
+Si je devais chercher les deux équations qui donnent directement la position  $B'$ en fonction de la position de $B$, le calcul (qui n'est pas à faire) serait fastidieux et le résultat complexe. Il donnerait :
+
+$\overline{S_2A'}=\frac{  n_2 \cdot\overline{S_2C_2} \cdot\left(\frac{ n \cdot \overline{S_1C_1} \cdot\overline{S_1A}}{(n-n_1) \cdot\overline{S_1A}+n_1 \cdot \overline{S_1C_1}}- \overline{S_1S_2}\right)}{(n_2n)\cdot\left(\frac{n\cdot\overline{S_1C_1}\cdot\overline{S_1A}}{(n-n_1)\cdot\overline{S_1A}\;+\;n_1\cdot\overline{S_1C_1}}-\overline{S_1S_2}\right)+n\cdot \overline{S_2C_2}}$   (équ.3a)
+
+$\overline{A'B'}=\overline{AB}\times\overline{M_{T-SO}}$, avec
+
+$\overline{M_{T-SO}}=\frac{n\cdot n_1\cdot \overline{S_1C_1}}{(n-n_1)\cdot\overline{S_1A}+n_1\cdot\overline{S_1C_1}}\:\times\:$
+$\frac{\overline{S_2C_2}}{(n2-n)\cdot\left(\frac{n\cdot\overline{S_1C_1}\cdot\overline{S_1A}}{(n-n_1)\cdot\overline{S_1A}+n_1\cdot\overline{S_1C_1}}-\overline{S_1S_2}\right)-n\cdot\overline{S_2C_2}}$    (équ.3b)
+
+Ces équations sont difficiles à établir et à retenir. Essayons au moins d'établir la relation de conjugaison de type $\dfrac{n_2}{\overline{S_2A'}}-\dfrac{n_1}{\overline{S_1A}}=\cdot\cdot\cdot$
+
+##### A la recherche d'une équation de conjugaison simple pour la lentille épaisse
+
+Les équations complexes 3a et 3b sont difficiles à manipuler. le plus simples est de repartir des équations (équ.1a) et (équ.2a) où les grandeurs $\dfrac{n_1}{\overline{S_1A}}$ et $\dfrac{n_2}{\overline{S_2A'}}$ apparaissent déjà. L'addition de chaque membre des équations (équ.1a) et (équ.2a) donne :
+
+$`\dfrac{n}{\overline{S_1A_{int}}}-\dfrac{n_1}{\overline{S_1A}}+\dfrac{n_2}{\overline{S_2A'}}-\dfrac{n}{\overline{S_2A_{int}}}$$=\dfrac{n-n_1}{\overline{S_1C_1}}+\dfrac{n_2-n}{\overline{S_2C_2}}`$
+
+En ne gardant au premier membre que les termes $\dfrac{n_1}{\overline{S_1A}}$ et $\dfrac{n_2}{\overline{S_2A'}}$ j'obtiens l'équation :
 
 $\dfrac{n_2}{\overline{S_2A'}}-\dfrac{n_1}{\overline{S_1A}}$$
 \:=\:\dfrac{n-n_1}{\overline{S_1C_1}}+\dfrac{n_2-n}{\overline{S_2C_2}}+\dfrac{n}{\overline{S_2A_{int}}}-\dfrac{n}{\overline{S_1A_{int}}}$
 
+Cette équation relativement simple semble convenir, mais c'est une *équation inutile*. Certes le membre de gauche ne contient que les informations sur les conditions d'utilisation de la lentille épaisse (_les indices_ $n_1$ _et_ $n_2$ _des milieux de part et d'autre de la lentille_) et de la position de l'objet $B$ (_la distance algébrique_ $\overline{S_1A_{int}}$_, position du point_ $A$ _par rapport au sommet_ $S_1$ _du premier dioptre rencontré par la lumière_). Mais pour être utile, le membre de droite ne devrait contenir que des grandeurs caractérisant la lentille épaisse elle-même : 
+* $n-n_1$ et $n-n_2$  : indices de réfraction différentiels entre matériau constituant la lentille et les milieux extérieurs.
+* $\overline{S_1C_1}$ et $\overline{S_2C_2}$ : rayons de courbures algébriques des faces d'entrée et de sortie de la lentille. 
+* $\overline{S_1S_2}$ épaisseur de la lentille.
+Or ce terme de droite contient aussi les distances algébriques $\overline{S_1A_{int}}$ et $\overline{S_2A_{int}}$ qui concernent la position de l'image intermédiaire $A_{int}$, or cette position dépend elle-même de la position du point objet initial $A$.
+
+Lorsque les **positions des points objets et images** sont **précisées par leur distances par rapport aux sommets $S_1$ et $S_2$**, frontières physiques de la lentille épaisse avec son axe optique, il n'existe **pas d'équation simple** séparant dans un terme de gauche les conditions d'utilisation de la lentille et de positions de l'objet et de l'image, et dans un terme de droite les seules caractéristiques des deux dioptres formant la lentille épaisse. Une formule de conjugaison générale pour tout système centré sera établie au chapitre "Etude des systèmes centrés".
+
+!! *POUR  ALLER  PLUS  LOIN* :<br>
+!!
+!! Tout système centré, qu'ils soit composé de deux ou de plusieurs éléments simples centrés sur un même axe optique, pourra être caractérisé par deux plans virtuels, appelés : 
+!! * plan principal objet (P) de point d'intersection avec l'axe optique $H$
+!! * plan principal image (P') de point d'intersection avec l'axe optique $H'$
+!! qui remplaceront respectivement la face d'entrée de la première lentille ou miroir du système par la lumière, et la face de sortie de la dernière lentille ou du dernier miroir).
+!!
+!! Ces plans permettront de définir une relation de conjugaison simple de forme connue : 
+!!
+!! $\frac{n'}{\overline{H'A'}}-\frac{n}{\overline{HA}}=V$ (avec V, vergence du système dans son environnement)
+!!
+!! et serviront de référence au positionnement des points focaux objet F et image F' du système dans son environnement :
+!!
+!! $V=-\frac{n}{\overline{HF}}=\frac{n'}{\overline{H'F'}}$
+!!
+!! Contrairement à $\overline{S_1S_2}$ toujours positive qu'elle remplacera, la distance algébrique $\overline{H_1H_2}$ pourra être positive ou négative.
+
+
+### Lentille mince 
+
+Une lentille est dite **lentille mince** lorsque la *distance entre les deux sommets $S_1$ et $S_2$* de la lentille est *petite devant chacun des rayons de courbures* des deux faces.
+
+Cette condition, $\overline{S_1S_2} \ll \overline{S_1C_1}$ et $\overline{S_1S_2} \ll \overline{S_2C_2}$ me permet de faire l'approximation $\overline{S_1S_2}\rightarrow 0$ dans les diverses équations de la lentille épaisse, considérant ainsi que les sommets $S_1$ et $S_2$ se confondent en un même point O.
+
+$\overline{S_1S_2}\rightarrow 0 \:\longrightarrow\:S_1=S_2=O$$\:\longrightarrow\:\overline{S_1C_1}\rightarrow\overline{OC_1}$ et $\overline{S_2C_2}\rightarrow \overline{OC_2}$
+ 
+##### Lentille mince en milieux extrêmes différents
+
+Je peux toujours considérer la lentille mince comme un système optique composé de deux dioptres sphériques centrés, et donc reprendre l'étude initiale de la lentille épaisse, mais avec l' approximation suivante : $S_1=S_2=O$ 
+
+
+##### Lentille mince plongé dans un même milieu
+
+Je peux toujours considérer la lentille mince comme un système optique composé de deux dioptres sphériques centrés, et donc reprendre l'étude initiale de la lentille épaisse, mais avec les approximations suivantes : $S_1=S_2=O$ et $n_1=n_2=n_{ext}$
+
+
+##### Lentille mince utilisés dans l'air ou dans le vide
+
+Ce sont les conditions d'utilisation des lentilles minces dans la très grande majorité des cas. Je reprendre l'étude initiale de la lentille épaisse avec les approximations suivantes :
+
+$S_1=S_2=O$ et $n_1=n_2=1$
+
+###### Pour le **premier dioptre** :
+
+La relation de conjugaison genérale $`\dfrac{n_{fin}}{\overline{SA_{ima}}}-\dfrac{n_{ini}}{\overline{SA_{obj}}}=\dfrac{n_{fin}-n_{ini}}{\overline{SC}}`$ et l'expression du grandissement transversal générale $`\overline{M_T}=\dfrac{n_{ini}\cdot\overline{SA_{ima}}}{n_{fin}\cdot\overline{SA_{obj}}}`$ des dioptres sphériques, donnent :
+
+* **$`\dfrac{n}{\overline{OA_{int}}}-\dfrac{1}{\overline{OA}}=\dfrac{n-1}{\overline{S_1C_1}}`$** (équ. 4a)
+
+* **$`\overline{M_{T-DS1}}=\dfrac{1\cdot\overline{OA_{int}}}{n\cdot\overline{OA}}`$** (équ. 4b)
+
+###### Pour le **second dioptre** :
+
+Ces mêmes expressions générales donnent :
+
+* **$`\dfrac{1}{\overline{OA'}}-\dfrac{n}{\overline{OA_{int}}}=\dfrac{1-n}{\overline{S_2C_2}}`$** (équ. 5a)
+
+* **$`\overline{M_{T-DS2}}=\dfrac{n\cdot\overline{OA'}}{1\cdot\overline{OA_{int}}}`$** (équ. 5b)
+
+Additionner entre elles les équations 1a et 2a donne:
+
+
+$\dfrac{n}{\overline{OA_{int}}}-\dfrac{1}{\overline{OA}}+\dfrac{1}{\overline{OA'}}-\dfrac{n}{\overline{OA_{int}}}=\dfrac{n-1}{\overline{S_1C_1}}+\dfrac{1-n}{\overline{S_2C_2}}$
+
+soit 
+
+**$\dfrac{1}{\overline{OA'}}-\dfrac{1}{\overline{OA}}=(n-1)\cdot\left(\dfrac{1}{\overline{S_1C_1}}-\dfrac{1}{\overline{S_2C_2}}\right)$**  (équ. 6)
+
+**Cette équation est vraiment utile** :
+* au premier membre ne se situent que les positions des points objet et image conjugués $A$ et $A'$  sur l'axe optique relativement au point O qui positionne la lentille mince sur cet axe.
+* au second membre n'intervient que ce qui caractérise la lentille mince (_ses deux rayons algébriques de courbure_ $\overline{S_1C_1}$, $\overline{S_2C_2}$ et l'indice de réfraction $n$ du matériau qui la compose (_la position du point image intermédiaire_ $A_{int}$ _de_ $A$ _par le premier dioptre n'intervient pas_) donc ce second membre est indépendant de la position du point-objet $A$ initial. Dans ces conditions, le premier membre de cette équation définit la vergence de la lentille mince.
+
+Connaissant les caractéristiques physiques ($\overline{S_1C_1}$, $\overline{S_2C_2}$, $n$) de la lentille, cette équation xxx *peut servir à calculer la position de tout point objet $A$ ou image $A'$ connaissant la position de son point conjugué* , c'est l'**équation de conjugaison avec origine au centre O de la lentille mince** lorsque les milieux de chaque côté de la lentille ont un même indice de réfraction $n=1$. On l'appelle *relation de conjugaison de Descartes*.
+
+Les expressions des distances focales objet et image de la lentille en fonction de ses caractéristiques physiques s'obtiennent facilement.
+
+* distance focale image $\overline{OF'}$  : $\left(|\overline{OA}|\rightarrow\infty\Rightarrow A'=F'\right)$
+
+$(équ. 7)\Longrightarrow\overline{OF'}=\dfrac{\overline{S_1C_1}\cdot\overline{S_2C_2}}{(n-1)\cdot(\overline{S_2C_2}-\overline{S1_C1})}$.
+
+* distance focale objet $\overline{OF}$  : $\left(|\overline{OA'}|\rightarrow\infty\Rightarrow A=F\right)$
+
+$(équ. 8)\Longrightarrow\overline{OF}=\dfrac{\overline{S_1C_1}\cdot\overline{S_2C_2}}{(n-1)\cdot(\overline{S_1C_1}-\overline{S_2C_2})}$.
+
+Je peux réécrire la vergence (premier membre de l'équation de conjugaison) en fonction des distances focales objet et image, et je reconnais bien l'**équation de conjugaison de la lentille mince plongée un milieu d'indice de réfraction 1** apprise au niveau "collines" :
+
+**$\dfrac{1}{\overline{OA'}}-\dfrac{1}{\overline{OA}}=V=-\dfrac{1}{\overline{OF}}=\dfrac{1}{\overline{OF'}}$**  (équ. 9a)
+
+! *REMARQUE * :<br>
+! Du point de vue de l'optique paraxiale, les plans focaux objet et image sont situés à même distance de part et d'autre de la lentille mince ($\overline(OF)=-\overline(OF')$).  La lentille mince est donc optiquement symétrique. Elle est caractérisée par sa distance focale image $\overline(OF')$. Cette distance focale $\overline(OF')$ est algébrique, une distance focale négative indique une lentille mince divergente, ne distance focale positive indique une lentille mince convergente, 
+
+<!--Le principe du retour inverse de la lumière indique que si la lumière change de sens, le point focal objet devient point focal image et vice versa.-->
+
+<!-- Ce "pour aller plus lojn" suivant est long et mal calibré. Ou il est trop court, ou il est trop long. peut-être le scinder en deux, la partie courte ici, et une partie plus longue en vidéo dans la partie "Pour aller plus lojn" (partie M)-->
+!! *POUR  ALLER  PLUS  LOIN* :<br>
+!!  Du point de vue de l'optique paraxiale, toutes les lentilles minces caractérisées par une même distance focale $\overline(OF')$ sont optiquement symétrique et équivalentes, qu'elles soient physiquement symétriques ou non, et quelques soient leurs matériaux constitutifs. Ainsi les lentilles sphériques suivantes :
+!! 1. biconvexe symétrique en verre crown  (PSK) : $n=1.63$ , $\overline{S_1C_1}=+50\;cm$ , $\overline{S_2C_2}=-50\;cm$.
+!! 2. biconvexe symétrique en verre flint (BaF) : $n=1.63$ , $\overline{S_1C_1}=+50\;cm$ , $\overline{S_2C_2}=-50\;cm$. 
+!! 3. plan-convexe en verre flint (BaF) : $n=1.63$, $\overline{S_1C_1}=19.3\;cm$ côté convexe.
+!!
+!! ont une même distance focale image $\overline(OF')=+30.5\;cm$.
+!!
+!! Pourtant leurs comportements optiques réels seront légèrement différents. L'écart entre le comportement optique réel et le comportement décrit par l'optique paraxiale est dit lié aux aberrations.
+!!
+!! * Aberration chromatique : L'indice de réfraction varie légèrement avec la longueur d'onde dans le domaine visible (_selon les types de matériaux, la variation de l'indice de réfraction limité au domaine visible est modélisé par différentes fonctions de la longueur d'onde : fonctions de Cauchy, de Briot, de  Sellmeier_). Ainsi, selon la loi de Snell-Descartes, un même rayon lumineux polychromatique incident sur un dioptre avec un angle non nul donnera lieu à différents rayons (_spectre de raies_) ou un faisceau de rayons (_spectre continu_) émergents monochromatiques : c'est le phénomène de dispersion chromatique. Ainsi la position des plans focaux objet et image varient continuement sur une petite plage de distance en fonction de la longueur donde. Même dans des conditions de Gauss idéalement réalisées, un point objet diffusant une lumière blanche (de spectre continu) ne donnera pas un point image blanc, mais une petite étenddue colorée aux couleurs de l'arce-en-ciel. Ce phénomène de dispersion est bien connu dans le cas d'un prisme qui décompose la lumière incidente en un faisceau coloré, mais ce phénomène est aussi présent lorsque la lumière traverse une lentille (_même si le résultat est moins accentué grâce à sa forme_). C'est le nombre d'Abbe qui caractérise ce phénomène de dispersion chromatique : plus il est petit plus le phénomène de dispersion est important.<br>
+!! Dans l'exemple, les lentilles 1 et 2 ont mêmes rayons de courbure et un même indice de réfraction, mais le lentille en verre flint (BaF) présentera une aberration chromatique beaucoup plus importante que la lentille en verre crown (PSK). 
+!!
+!! * L'aberration géométrique : xxx.<br>
+!! Pour une lentille plan-convexe, l'aberration géométrique sera différente selon le sens de traversée de la lentille par la lumière. Les lentilles 2 et 3 sont réalisées dans un même verre et sont caractérisées par une même distance focale image, elles se comportent de façons identiques selon l'optique paraxiale. Cependant, éclairées par un même faisceau monochromatique (_pour éviter l'aberration chromatique_) sous incidence normale, c'est la lentille plan-convexe utilisée avec la lumière incidente
+
+Le grandissement transversal de la lentille mince est le produit des grandissements transversaux de chacun des deux dioptres qui composent la lentille mince. En effet :
+
+$M_T  =\dfrac{\overline{A'B'}}{\overline{AB}}=\dfrac{\overline{A_{int}B_{int}}}{\overline{AB}}\cdot\dfrac{\overline{A'B'}}{\overline{A_{int}B_{int}}}=M_{T-DS1}\cdot M_{T-DS2}$
+
+Le calcul de son expression :
+
+$M_{T-DS1}\cdot M_{T-DS2}$$=\dfrac{1\cdot\overline{OA_{int}}}{n\cdot\overline{OA}}\times\dfrac{n\cdot\overline{OA'}}{1\cdot\overline{OA_{int}}}$
+
+se simplifie en :
+
+**$M_{T-thinlens}=\dfrac{\overline{OA'}}{\overline{OA}}$**    (équ. 9b)
+
+Les équations 9a et 9b prennent le point O, centre de la lentille mince, pour référence des distances algébriques $\overline{OA}$, $\overline{OA'}$, et permettent de calculer les distances focales objet $\overline{OF}$ et image $\overline{OF'}$, et donc de postionner les points focaux $F$ et $F'$ de la lentille mince.
+
+Si c'est points sont déjà connus, alors je peux déduire une autre formule de conjugaison et une autre expression du grandissement transversal en prenant les points focaux $F$ et $F'$ pour références des distances algébriques :
+
+$\dfrac{1}{\overline{OA'}}-\dfrac{1}{\overline{OA}}=V=-\dfrac{1}{\overline{OF}}=\dfrac{1}{\overline{OF'}}$
+
+Je peux par exemple multiplier chaque membre de l'équation par  $\overline{OA}$
+
+$\dfrac{\overline{OA}}{\overline{OA'}}-1=\dfrac{\overline{OA}}{\overline{OF'}}$
+
+Puis réduire au même dénominateur le membre de gauche
+
+$\dfrac{\overline{OA}-\overline{OA'}}{\overline{OA'}}=\dfrac{\overline{OA}}{\overline{OF'}}$
+
+Je fais le produit en croix
+
+$\overline{OA'}\cdot\overline{OA}=(\overline{OA}-\overline{OA'})\cdot\overline{OF'}$
+
+Je fais apparaître $\overline{FA}$ et $\overline{F'A'}$
+
+$(\overline{OF'}+\overline{F'A'})\cdot(\overline{OF}+\overline{FA})$$=[(\overline{OF}+\overline{FA})-(\overline{OF'}+\overline{F'A'})]\cdot\overline{OF'}$
+
+Je ne garde par exemple que la distance focale image en remplaçant $\overline{OF}$ par $-\overline{OF'}$
+
+$(\overline{OF'}+\overline{F'A'})\cdot(-\overline{OF'}+\overline{FA})$$=(-\overline{OF'}+\overline{FA}-\overline{OF'}-\overline{F'A'})\cdot\overline{OF'}$
+
+J'effectue les produits
+
+$-\overline{OF'}^2+\overline{OF'}\cdot\overline{FA}-\overline{OF'}\cdot\overline{F'A'}+\overline{FA}\cdot\overline{F'A'}$$=-\overline{OF'}^2+\overline{OF'}\cdot\overline{FA}-\overline{OF'}^2-\overline{OF'}\cdot\overline{F'A'}$
+
+et je simplifie
+
+**$\overline{FA}\cdot\overline{F'A'}=-\overline{OF'}^2$**
+
+**Cette équation est vraiment utile** :
+* au premier membre ne se situent que les positions des points objet et image conjugués $A$ et $A'$  sur l'axe optique relativement aux foyers objet $F$ et image $F'$ de la lentille mince avec de chaque côté un milieu d'indice de réfraction $n=1$.
+* au second membre n'intervient que ce qui caractérise la lentille mince dans ces conditions d'utilisation (_ses distances algébriques _ $\overline{OF}$ et  $\overline{OF'}$ , positions des points focaux objet et image par rapport au point O). 
+
+Connaissant la distance focale image $\overline{OF' }$de la lentille (_attention, comme toute distance en optique géométrique, cette distance est algébrique_), cette équation xxx *peut servir à calculer la position de tout point objet $A$ ou image $A'$ connaissant la position de son point conjugué* , c'est l'**équation de conjugaison de la lentille mince, avec origines aux foyers**,
+lorsque les milieux de chaque côté de la lentille ont un même indice de réfraction $n=1$. On l'appelle *relation de conjugaison de Newton*.
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
 
 
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@@ -1,5 +1,150 @@
 ---
-title: 'new course  : overview'
+title: lens-overview
+media_order: 'lens-convergent-N3-en.jpeg,lens-divergent-N3-en.jpeg,Const_lens_conv_point_AavantF2.gif,thick-lens-water-air.gif,Lentille_epaisse_Gauss_incl_v1.gif,2-centered-refracting-surfaces-1-all.gif,2-refracting-surface-physical-system.jpeg,2-centered-refracting-surfaces-direction-axis.gif,Lentille_epaisse_principe_ok.gif,lentille_relle_representation_v1.gif,Const_lens_conv_point_AapresO.gif,lens-convergent-N3-es.jpeg,lens-convergent-N3-fr.jpeg,lens-divergent-N3-es.jpeg,lens-divergent-N3-fr.jpeg'
 ---
 
-new course  : overview
\ No newline at end of file
+### Two successive and centered spherical refracting surfaces
+
+!!! *WHERE YOU ARE* :<br>
+!!! Observation $\Rightarrow$ Geometrical optic interpretation $\Rightarrow$ Fermat's Principle $\Rightarrow$ The 5 optical laws $\Rightarrow$ Paraxial approximation $\Rightarrow$ Simple optical element $\Rightarrow$ System of 2 simple optical elements.
+
+! *THIS CHAPTER AIMS AT* :
+! * Deeply understand and better master thin lenses.<br>
+! * Understand when the lens equation and the coresponding transverse magnification expression can be used, and when they are not correct.
+!  * Understand need and requirement of new concepts to master esaily and efficientlynext main chapter "Centered optical systems".
+
+*Two successive and centered spherical refracting surfaces* = **thick lens**
+
+##### Thick lens as a physical system
+
+**Physical system** =  *spatial distribution of the refracting indexes values* (_variations of refracting index can be discontinuous with interfaces (_refracting surfaces, lenses, mirrors_) or continuous (graded-index optical fiber)_.
+
+**Optical system** = *oriented physical system* = *physical systems + bodies (1) + a direction (2)* <br>
+* (1) : which emit, diffuse or reflect the ambiant light.
+* (2) : direction of light propagation considered through the physical system. 
+
+**Difference** between physical and optical system in optics :
+
+Example of the lensball :
+Physical system of a lensball :
+
+**Thick lens** physical system :<br>
+Most general : *3 different transparent media with their own refractive index values*, and *2 local spherical interfaces* that separate these media, and *centered on the straigth line* that joins their centers of curvature._
+
+**Examples** in images :
+
+![](2-refracting-surface-physical-system.jpeg)
+
+##### Thick lens as an optical system
+
+**Optical system** = *oriented physical system* = *physical systems + bodies (1) + a direction (2)* 
+
+* (1) : which emit, diffuse or reflec the ambiant light
+* (2) : direction of light propagation considered through the two refracting surfaces. 
+
+*From physical system to optical system* : **a scenario to build** :
+* Where is the object that is imaged ?
+* In what direction are we searching for images ?
+* what are the reflecting or refracting interfaces we take into account.
+
+<!--To define an optical system, you have to find a scenario : where is the objet to be imaged or viewed ? And where is the real image of the object to be registered by a matrix sensor or where is located the eye of the observator ? This gives you the direction of propagation (from object to real image or eye) through the optical system. This direction of propagation is part of the description of the optical system. In the figure above the optical systems are each time two ordered spherical refracting surfaces._-->
+
+
+_The physical system consists of two bubble aquariums side by side. In each of them, a fish, and the two fish, Jones and Tessa face each other. These two situations correspond to two optical systems: "Tessa looks at Jones" and "Jones looks at Tessa" (the order of crossing of the refracting surfaces by the light is reversed in both cases). In the situtation we want to describe, the direction of the light is indicated (the brown arrow in the figures)_
+
+**Graphical representation** (drawing) and **analytical representation** (*3 algebraic distances* : 2 radius of curvature $\overline{S_1C_1}$, $\overline{S_2C_2}$,+ distance between the two vertices of the refracting surfaces $\overline{S_1S_2}$ *when used in the paraxial (or same, gaussian) approximation, so when considered in paraxial optics.
+
+_ In order to identify conjugated points, to construct the final image of a specific object for example, the optical axis of the optical system is plotted, vertices and centers of curvature of spherical refracting surfaces are localised on the optical axis. Because 
+
+
+
+!!! Thick lens
+
+
+
+
+![](thick-lens-water-air.gif)
+
+![](2-centered-refracting-surfaces-1-all.gif)
+
+![](2-centered-refracting-surfaces-direction-axis.gif)
+
+![](thick-lens-water-air.gif)
+
+![](Const_lens_conv_point_AapresO.gif)
+
+![](Const_lens_conv_point_AavantF2.gif)
+
+![](Lentille_epaisse_Gauss_incl_v1.gif)
+
+![](thick-lens-water-air.gif)
+
+![](Lentille_epaisse_principe_ok.gif)
+
+![](lentille_relle_representation_v1.gif)
+
+
+### The thin lens
+
+##### Objective
+to **focuse or disperse the light**,<br> 
+with often the final goal, alone or as part of optical instruments, to **realize images**.
+
+##### Physical principle
+**uses the refractive phenomenon**, described by the Snell-Descartes' law.
+
+##### Characterization of its efficiency
+(efficiency to realize its objective)
+
+**Vergence** = **dioptric power** V of the lens :
+
+* **unit** : in S.I. : the *diopter*, of symbol $\delta$<br>
+ 1 diopter = 1 $\delta$ = 1 $m^{-1}$).
+* **positive vergence** ($V>0)\:\Longleftrightarrow$ *light focalisation : convergent lens*.<br>
+* **negative vergence** ($V<0)\:\Longleftrightarrow$ *light dispersion : divergent lens*.<br>
+* **absolute value** of the vergence ($|V|$) : *increases as the optical phenomenon (focalisation or dispersion) increases* (as the corresponding deviation of light rays increases).<br>
+* **interest** : The *total vergence* of several __contiguous thin lenses__ is the *sum of the vergences of each of the lenses* : $V=\sum V_i$.
+
+or (equivalent)
+
+**image focal length** $f'$ of the lens : 
+
+* **positive $f'$** ($f'>0)\:\Longleftrightarrow$ *focuses light : convergent lens*<br>
+* **negative $f'$** ($f'<0)\:\Longleftrightarrow$ *disperse light : divergent lens*<br>
+* **absolute value of $f'$** ($|f'|$) : *decreases as the optical phenomenon (focalisation or dispersion) increases*.<br>
+* **interest** : For thin lenses, the **algebraic value of $f'$** give the *position of the plane* (perpendicular to the optical axis) and from the lens center *where the image of an object at infinity takes place*. 
+
+! Nearby in all application, same medium (same refractive index) in both sides of the lens :<br>
+! $\Longrightarrow$ object focal lenght $f$ is the opposite of image focal length $f'$ : $f=-f'$<br>
+! $\Longrightarrow$ only absolute value $|f'|$ of $f'$ is given, and the lens is specified to be convergeng or divergent.
+
+**Relation between vergence (dioptric power), image and object focal lengthes**
+
+if the refractive index $n_{ini}$ of the medium in which the incident light on the lens propagates, and $n_{fin}$ of the medium in which the light emerges from the lens, then :
+
+$V=-\dfrac{n_{ini}}{f}=+\dfrac{n_{fin}}{f'}$
+
+##### Constitution
+
+Piece of **glass, quartz, plastic** (for visible and near infrared and UV).<br>
+**Rotationally symmetrical**,<br>
+**Thin**,<br>
+**2 polished surfaces** perpendicular to its axis of symmetry, **either or both curved** (and most often spherical).
+
+##### Classification of thin lenses
+
+**Convergent lenses** = **positive lenses** 
+
+![](lens-convergent-N3-en.jpeg)
+
+**Divergent lenses** = **negative lenses** 
+
+![](lens-divergent-N3-en.jpeg)
+
+### Brief chronology
+
+### Modeling a lens
+
+##### 
+
+
diff --git a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/02.paraxial-optics-simple-elements/03.lens/02.lens-overview/cheatsheet.fr.md b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/02.paraxial-optics-simple-elements/03.lens/02.lens-overview/cheatsheet.fr.md
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+++ b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/02.paraxial-optics-simple-elements/03.lens/02.lens-overview/cheatsheet.fr.md
@@ -1,9 +1,24 @@
 ---
 title: 'nouveau cours : synthèse'
-media_order: '2-centered-refracting-surfaces-direction-axis.gif,2-centered-refracting-surfaces-1-all.gif,lentille_relle_representation_v1.gif,Const_lens_conv_point_AapresO.gif,Const_lens_conv_point_AavantF2.gif,Lentille_epaisse_principe_ok.gif,Lentille_epaisse_Gauss_incl_v1.gif'
+media_order: 'lens-divergent-N3-fr.jpeg,lens-convergent-N3-fr.jpeg,2-centered-refracting-surfaces-direction-axis.gif,2-centered-refracting-surfaces-1-all.gif,lentille_relle_representation_v1.gif,Const_lens_conv_point_AapresO.gif,Const_lens_conv_point_AavantF2.gif,Lentille_epaisse_principe_ok.gif,Lentille_epaisse_Gauss_incl_v1.gif,2-refracting-surface-physical-system.jpeg,lens-convergent-N3-en.jpeg,lens-convergent-N3-es.jpeg,lens-divergent-N3-en.jpeg,lens-divergent-N3-es.jpeg,thick-lens-water-air.gif'
 ---
 
-nouveau cours : synthèse
+### Classification des lentilles minces
+
+**Lentilles convergentes** =  **lentilles positives**
+
+_Lentilles "positives" car leurs distances focales images algébriques sont positives (l'axe optique étant orienté positivement dans le sens de propagation de la lumière)_
+
+![](lens-convergent-N3-fr.jpeg)
+
+**Lentilles divergentes** =  **lentilles négatives**
+
+_Lentilles "positives" car leurs distances focales images algébriques sont positives (l'axe optique étant orienté positivement dans le sens de propagation de la lumière)_
+
+![](lens-divergent-N3-fr.jpeg)
+
+
+`nouveau cours : synthèse
 
 ![](Const_lens_conv_point_AapresO.gif)
 
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@@ -1,31 +1,165 @@
 ---
 title: 'new course : beyond'
+media_order: 'lentille-boule-orleans-1bis.jpg,lensball-brut-820-760.jpg,Einstein-ring-free.jpg'
 ---
 
-! *YOUR  CHALLENGE*  :  look at the picture, and think of the right answers to the following questions
+! *YOUR  CHALLENGE*  :  Looking at a cathedral through a lensball. Can you predict your observation?
 !
-! _Do not look at the answer, take time to think, a few days if necessary. The time to build your mental representation of the phenomenon, to formulate it in words is important, a thousand times more important than the ephemeral instant where you read the fews words of the solution._
+! ![](lensball-brut-820-760.jpg)
 !
+! *Discovery time : 2 hours*<br>
+! *Resolution time : 30 minutes*
 ! <details markdown=1>
-! <summary>
-! INDICE "key word"
+!<summary>
+! I choose it
 ! </summary>
-! diffusion
-! </details>
+! A lensball is a polished spherical ball of radius $R=5 cm$, made of glass of refractive index value $n_{glass}=1.5$. The cathedral is 90 meters high, and you stand with the lensball 400 meters from the cathedral. You look at the cathedral through the lens, your eye being at 20 cm from the center of the lens. What would you expect to see? 
+!
+! * _The resolution time is the typical expected time to be allocated to this problem if it was part of an examen for an optics certificate_. 
+!
+! * _The discovery time is the expected time you require to prepare this challenge if you don't have practice. However, this is just an indication, take as much time as you need. The time to question yourself serenely about how to handle the problem, about the method of resolution and its validity, about some possible approximations if they can be justified, and the time  you need to check the equations if you have not previously memorised them and to perfom the calculation, are important._
+!
 ! <details markdown=1>
 ! <summary>
-! ANSWER
+! Ready to answer M3P2 team questions ?
+! </summary>
+!<!--question 1-->
+!<details markdown=1>
+!<summary>
+! What is the scientifical framework you choose to study this problem ?
+!</summary>
+! * All the characteristic sizes in this problem are much bigger than the wavelength of the visible radiation ($\lambda\approx5\mu m$), so I deal with this problem in the framework of geometrical optics, and in the paraxial approximation in order to characterize the image.
+!
+! * The cathedral sustains an angle of $arctan\dfrac{90}{400}=13°$ from the lensball. This value seems reasonable to justify at first order the use of the paraxial approximation (_we usually consider that angles of incidence would not exceed 10° on the various simple optical element encountered between the objet (here the cathedral) and the final image (retina of the eye or matrix sensor of a camera_).
+!</details>
+!<!--question 2-->
+!<details markdown=1>
+!<summary>
+! Describe the optical system for this use of the lensball.
+!</summary>
+! * The lensball breaks down into two refracting spherical surfaces sharing the same centre of curvature C and of opposite radius (in algebraic values).
+!</details>
+!<!--question 3-->
+!<details markdown=1>
+!<summary>
+! What is your method of resolution ?
+!</summary>
+! * You don't use general equations 3a and 3b for a thick lens, they are too complicated to remind, and you don't have in m3p2 to "use" but to "build a reasoning". And you don't know at this step how to handle with centered optical systems. 
+!
+! * But this system is simple, so you will calculate the image of the cathedral by the first spherical refracting surface $DS_1$ encountered by the light from the cathedral $DS_1$. Then this image becomes the object for the second  spherical refracting surface $DS_2$ and so I can determine position and size of the final image.
+!
+! * For a spherical refracting surface, general equations are :<br><br>
+! $`\dfrac{n_{fin}}{\overline{SA_{ima}}}-\dfrac{n_{ini}}{\overline{SA_{obj}}}=\dfrac{n_{fin}-n_{ini}}{\overline{SC}}`$ for the position.<br>
+! $`\overline{M_T}=\dfrac{n_{ini}\cdot\overline{SA_{ima}}}{n_{fin}\cdot\overline{SA_{obj}}}`$ for the transverse magnification.
+!</details>
+!<!--question 4-->
+!<details markdown=1>
+!<summary>
+! How do you set down your calculations?
 ! </summary>
-! diffusion
+! * The optical axis is the straight line that joins the center C of the lens to my eye, positively oriented in the direction of light propagation light for that observation, so from the cathedral to my eye. 
+! * First spherical refrating surface $DS1$ : $S_1C_1=+5\:cm$, $n_{ini}=1$ (air) and $n_{fin}=1.5$ (glass).<br>
+!  Second spherical refrating surface $DS2$ : $S_1C_1=-5\:cm$, $n_{ini}=1.5$ (glass) and $n_{fin}=1$ (air)<br>
+!  Distance between $DS1$ and $DS2$ vertices : $S_1S_2=+10\:cm$<br>
+!  Object cathedral $AB$ :  $AB=90\;m$ and $S_1A=-400\;m$<br>
+!  Let us write $A_1B_1$ the intermediate image (the image of the cathedral given by $DS1$.
+!
+!  * Specific equations for $DS1$ are :<br><br> 
+! $`\dfrac{1.5}{\overline{S_1A_1}}-\dfrac{1}{\overline{S_1A}}=\dfrac{0.5}{\overline{S_1C_1}}`$ (équ. DS1a),  and 
+! $`\overline{M_T}=\dfrac{\overline{S_1A_1}}{1.5\cdot\overline{S_1A}}`$ (équ. DS1b)<br><br>
+! Specific equations for $DS2$ are :<br><br> 
+! $`\dfrac{1}{\overline{S_2A'}}-\dfrac{1.5}{\overline{S_2A_1}}=-\dfrac{0.5}{\overline{S_2C_2}}`$  (équ. DS2a),  and 
+! $`\overline{M_T}=\dfrac{1.5\cdot\overline{S_2A'}}{\overline{S_2A_1}}`$ (équ. DS2b)<br><br>
+! The missing link between these two sets of equations is :<br>
+! $\overline{S_2A_1}=\overline{S_2S_1}+\overline{S_1A_1}=\overline{S_1A_1}-\overline{S_1S_2}$.
+!</details>
+!<!--question 5-->
+!<details markdown=1>
+!<summary>
+! Do you see some approximation that can be done ? 
+!</summary>
+! * In the visible range, refractive index values of transparent material are in the range [1 ; 2], then the focal lengthes of a spherical refractive surface (object as well as image) are expected to remain in the same order of magnitude than the radius of curvature, so a few centimeters in this case (we talk in absolute value here). 
+!
+! * We can if we want just check this fact for $DS1$ ($|S_1C_1|=5\;cm$) using équation DS1 :<br>
+! \- considering $\overline{S_1A_1}\longrightarrow\infty$ to obtain the object focal length $\overline{S_1F_1}$} we get :<br>
+! $`-\dfrac{1}{\overline{S_1F_1}}=\dfrac{0.5}{\overline{S_1C_1}}$ $\Longrightarrow=\overline{S_1F_1}=-10\;cm`$<br><br>
+! \- considering $`\overline{S_1A}\longrightarrow\infty`$ to obtain the image focal length $`\overline{S_1F'_1}`$ we get :<br>
+! $`\dfrac{1.5}{\overline{S_1F'_1}}=\dfrac{0.5}{\overline{S_1C_1}}\Longrightarrow\overline{S_1F'_1}=+15\;cm`$
+! 
+! * The distance of the cathedral from the lensball $|\overline{S_1A}|=90\;m$ is huge compared to the object focal length $|\overline{S_1F_1}|=10\;cm`$, we can consider that the cathedral is at infinity from the lensball and so the image $A_1B_1$ of the cathedral stands quasi in the image focal plane of $DS1$ : $\overline {S_1A_1}=\overline {S_1F'_1}=+15cm$. So we can directly use equation DS2a with :<br>
+! $\overline{S_2A_1}=\overline{S_2F'_1}=\overline{S_2S_1}+\overline{S_1F'_1}$ $=\overline{S_1F'_1}-\overline{S_1S_2}=+15-10=+5\;cm$..
+!</details>
+!
+!<!--question 6-->
+!<details markdown=1>
+!<summary>
+! Where is the image and how tall it is ?
+!</summary>
+! * To perform calculation, you must choose a unic lenght unit in your calculation, here $cm$ or $m$. We choose $m$ below.
+! * Equation DS1a gives :<br>
+! $\dfrac{1.5}{\overline{S_1A_1}}-\dfrac{1}{-400}=\dfrac{0.5}{0.05}$ $\Longrightarrow\overline{S_1A_1}=0.15\;m$<br>
+! With more than 2 significant figures, your calculator would tell you $0.150037$, which nearly exactly the value of $\overline{S_1F'_1}=+0.15\;m$, so the approximation $\overline{S_1A_1}=\overline{S_1F'_1}$ you could have done is fully justified.
+! 
+! * Equation DS2a gives :<br>
+! $\dfrac{1}{\overline{S_2A'}}-\dfrac{1.5}{-0.1+0.15}=\dfrac{-0.5}{-0.05}$ $\Longrightarrow\overline{S_2A'}=0.025\;m$
+!
+! * The final image is real, and stands 2.5 cm in front of the lensball in the side of your eye. Do not bring your eye or camera too close of the lensball \!
+!
+! * The size of an image (transversally to the optical axis) is given by the transverse magnification $M_T$.  By Definition $M_T$ is the ratio of the algebraic size of the final image $\overline{A'B'}$  to the algebraic size of the initial object $\overline{AB}$. With an intermediate image, it can be break down :<br><br>
+!  $M_T=\dfrac{\overline{A'B'}}{\overline{AB}}$ $=\dfrac{\overline{A'B'}}{\overline{A_1B_1}}\times\dfrac{A_1B_1}{\overline{AB}}$<br><br>
+! It is the product of the transverse magnifications of the cathedral introduced by the two spherical refracting surfaces of the lensball. <br><br>
+! $\overline{M_T}$ introduced by $DS1$ is 
+! $`\overline{M_T}=\dfrac{\overline{S_1A_1}}{1.5\cdot\overline{S_1A}}`$ $=\dfrac{+0.15}{1.5\times(-400)}=-0.00025$<br><br>
+! $\overline{M_T}$ introduced by $DS2$ is 
+! $\overline{M_T}=\dfrac{1.5\cdot\overline{S_2A'}}{\overline{S_2A_1}}$ $=\dfrac{1.5\cdot\overline{S_2A'}}{\overline{S_1A_1}-\overline{S_1S_2}}$ $=\dfrac{1.5\cdot0.025}{+0.15-0.10} =0.75$<br><br>
+! So $\overline{M_T}$ introduced by the lensball is :<br><br>
+! $\overline{M_T}=-0.00025\times0.75$ $=-0.00019\approx-1.9\cdot10^{-4}$<br><br>
+! The image is  $\dfrac{1}{-1.9\cdot10^{-4}}\approx5300$ smaller than the cathedral.<br><br> 
+! $M_T=\dfrac{\overline{A'B'}}{\overline{AB}}\approx8\cdot10^{-4}$ $\Longrightarrow\overline{A'B'}=\overline{AB} \times M_T $ $=1.9\cdot10^{-4} \times 90\;m=-0.017\;m$<br><br>
+! The image is 1.7 cm height and it is reversed.
+!</details>
+!
+!<!--question 7-->
+!<details markdown=1>
+!<summary>
+! What is the apparent magnification of the cathedral ?
+!</summary>
+! 
+! * "apparent magnification" = "angular magnification" = "magnifying power".
+!
+! * As calculated previously, standing 400 metres from the cathedral, the 90 m heigh cathedral sustends the apparent angles of $\alpha=arctan\left(\dfrac{90}{400}\right)=0.221\;rad=12.7°$ at your eye.
+!
+! * The image of the cathedral is 1.7 cm heigth and is located between the lens (from its vertex $S2$) and your eyes and at 2.5cm from the lens. If your eye is 20cm away from the lens, so the distance eye-image is 17.5 cm (we use no algebraic values). Thus the image of the catedral subtends the apparent angle $\alpha'=arctan\left(\dfrac{1.7}{17.5}\right)=0.097\;rad=5.6°$ at your eye.
+!
+! * The apparent magnification $M_A$ of the cathedral throught the lensball for my eye in that position is<br>
+!  $M_A=\dfrac{\alpha'}{\alpha}=\dfrac{0.097}{0.221}=0.44$.<br><br>
+!  Taking into account that the image is reversed, the algebraic value of the apparent magnification is $\overline{M_A}=-0.44$.
+!
+! * You could obtained directly this algebraic value of $M_A$ by considering algebraic lengthes and angles values in the calculations :<br><br> $\overline{M_A}=\dfrac{\overline{\alpha'}}{\overline{\alpha}}$ $=\dfrac  {arctan\left(\frac{-0.017}{-0.175}\right)} {arctan\left(\frac{90}{-400}\right)}$ $=\dfrac{0.097}{-0.221}=-0.44$
+!  </details>
+!
+! ![](lentille-boule-orleans-1bis.jpg)<br>
+! _Cathedral of Orleans (France)_
+!
+! </details>
+! </details>
 ! </details>
 
+!! *BEYOND* : The gravitationnal lensball (or Einstein's ring), due to a black hole or a galaxy.
+!! Similarities, and differences.
+!!
+!! ![](Einstein-ring-free.jpg)
+!!
+!! <details  markdown=1>
+!! <summary>
+!! To see
+!! </summary>
+!! still to be done, in progress. 
+!! </details>
 
 !!!! *DIFFICULT POINT* (contribute, or indicate a difficult point of understanding)
 !!!!
 
-!! *BEYOND* (to contribute)
-!!
-
 !! *CULTURAL POINT* (contributor)
 !!
 
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+---
+title: 'new course : beyond'
+media_order: 'syst-physique-syst-optique.gif,lentille-boule-transparence.jpg,lentille-boule-2-reflexions.jpg,lentille-boule-seule.jpg'
+---
+
+! *YOUR  CHALLENGE*  :  An object (a painting), a physical system (a lensball), how many scenarios and optical systems?
+!
+! ![](syst-physique-syst-optique.gif)
+!
+! *Discovery time : 30 minutes*<br>
+! *Resolution time : 10 minutes*
+! <details markdown=1><!--1-->
+!<summary>
+! I choose it
+!
+! </summary>
+! A lensball is a simple physical system: a sphere of glass of refractive index $n=1.5$ and of radius $R=5\;cm$.
+!
+! A ball lensball is placed in front of a painting. Depending on the position of the observer or the camera, the optical system (the sequence of simple optical elements crossed by light between the physical object and the observed image) that forms the image differs.
+!
+! Observe the 3 images of the painting given by the lensball :
+!
+! Image 1
+! ![](lentille-boule-transparence.jpg)
+! Images 2 (the smallest) and 3
+! ![](lentille-boule-2-reflexions.jpg)
+!
+! For each image of the painting, can you identify the optical system, then specify the characteristics of the various simple elements that constitute the system and their relative distances?
+!
+! * _The resolution time is the typical expected time to be allocated to this problem if it was part of an examen for an optics certificate._ 
+! * _The discovery time is the expected time required to prepare this challenge if you don't have practice. But take as much time as you need._
+!
+!<\details>
+! <details markdown=1>
+! <summary>
+! Ready to answer M3P2 team questions for image 1?
+! </summary>
+!
+! <details markdown=1>
+! <summary>
+! Where is the painting located?
+! </summary>
+! * The painting is located on the other side of the lens, in relation to you.
+! </details>
+! <details markdown=1>
+! <summary>
+! What is the optical system giving the image of the painting?
+! </summary>
+! * The optical system is composed of two spherical refracting surfaces, centered on the same optical axis.
+! </details>
+! <details markdown=1>
+! <summary>
+! How do you characterize each of the single optical elements that make up this optical system, and their relative distances?
+! </summary>
+! * The optical axis is oriented positively in the direction of light propagation (from the painting towards the lensball).
+!
+! * The first spherical refracting surface $DS1$ encountered by the light has the follwing characteristics :<br>
+! $\overline{S_1C_1}=+|R|=+5\;cm$ , $n_{ini}=1$ and $n_{fin}=1.5$
+!
+! * The second spherical refracting surface $DS2$ encountered by the light has the follwing characteristics :<br>
+! $\overline{S_2C_2}=-|R|=-5\;cm$ , $n_{ini}=1.5$ and $n_{fin}=1$
+!
+! * Algebraic distance between $DS1$ and $DS2$ is : $\overline{S_1S_2}=+10\;cm$
+! </details>
+! <details markdown=1>
+! <summary>
+! If you had to determine the characteristics of the image (position, size), how would you handle the problem?
+! </summary>
+! * $DS1$ gives an image $B_1$ of an object $B$. This image $B_1$ for $DS1$ becomes the object for $DS2$. $DS2$ gives an image $B'1$ of the object $B_1$
+! </details>
+! </details>
+! <!--FOR IMAGES 2 & 3-->
+!
+! <details markdown=1>
+! <summary>
+!
+! Ready to answer M3P2 team questions for images 2 and 3?
+! </summary>
+!
+! <details markdown=1>
+! <summary>
+! Where is the painting located?
+! </summary>
+! * The painting is located on the same side of the lens as you, behind you.
+! </details>
+! <details markdown=1>
+! <summary>
+! What are the two optical systems at the origin of the two images of the painting? And can you characterize each of the single optical elements (+ their relative distances) that make up each of these optical systems ?
+! </summary>
+! * A first optical system $OS1$ is composed of a simple convexe spherical mirror (the object is reflected on the front face of the ball lensball). Keaping the ioptical axis positively oriented in the direction of the incident light propagation on the lensball, the algebraic value of the mirror radius is : $\overline{SC}=+5\;cm$.
+! * The second optical system $OS2$ is composed of three simple optical elements :<br><br>
+! 1) The light crosses a spherical refracting surface $DS1$ with characteristics : $\overline{S_1C_1}=+|R|=+5\;cm$ , $n_{ini}=1$ and $n_{fin}=1.5$.<br><br>
+! 2) Then the light is reflected at the surface of the last lensball interface that acts like a spherical mirror of characteristics : $\overline{S_2C_2}=-|R|=-5\;cm$, $n=1.5$.<br><br>
+! 3) Finally the light crosses back the first interface of the lensball that acts like a spherical refracting surface those characteristics are : $\overline{S_3C_3}=+|R|=+5\;cm$ , $n_{ini}=1.5$ and $n_{fin}=1$.<br><br>
+! Relative algebraic distances between the different elements of $OS2$ are :<br>
+! $\overline{S_1S_2}=+10\;cm$ and $\overline{S_2S_3}=-10\;cm$
+! </details>
+! <details markdown=1>
+! <summary>
+! Which image is associated with each of the optical systems?
+! </summary>
+! * It is difficult to be 100% sure before having made the calculations. 
+! </details>
+! <details markdown=1>
+! <summary>
+! Why do we had to take the picture in the darkness, with only the painting illuminated behind the camera, to obtain images 2 and 3 ?
+! </summary>
+! * At a refracting interface, part of the light incident power is refracted, and part is reflected. For transparent material like glass and for visible light, the part of the reflected power is small. If the room had been homogeneously illuminated, the images 2 and 3 of the painting on the wall behind the camera would have been faintly visible compared to the image of the front wall through the lensball.
+! </details>
+! </details>
+! </details>
+
+
+!!!! *DIFFICULT POINT* (contribute, or indicate a difficult point of understanding)
+!!!!
+
+!! *BEYOND* (to contribute)
+!!
+
+!! *CULTURAL POINT* (contributor)
+!!
+
+!!! *DO YOU MASTER ?*
+!!! <details markdown = 1>
+!!! <summary>
+!!! Describe the test.
+!!! </summary>
+!!! Texte du test (et ses images, figures, audio, video, etc ...)
+!!! ! [a figure for the test] (file-name)
+!!!
+!!! Texte de la question
+!!! <details markdown=1>
+!!! <summary>
+!!! Choix de réponse 1
+!!! </summary>
+!!! Texte si la réponse 1 est choisie
+!!! </details>
+!!! <details markdown=1>
+!!! <summary>
+!!! Choix de réponse 2
+!!! </summary>
+!!! Texte si la réponse 2 est choisie
+!!! </details>
+!!! <!--possibility to add answers, questions-->
+!!! -----
+!!! <details markdown=1>
+!!! <summary>
+!!! Solution complète
+!!! </summary>
+!!! Texte de la solution
+!!! </details>
diff --git a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/03.physical-system-versus-optical-system/03.new-course-beyond/annex.es.md b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/03.physical-system-versus-optical-system/03.new-course-beyond/annex.es.md
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+++ b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/03.physical-system-versus-optical-system/03.new-course-beyond/annex.es.md
@@ -0,0 +1,153 @@
+---
+title: 'nouveau cours : au-delà'
+media_order: 'lentille-boule-2-reflexions.jpg,lentille-boule-transparence.jpg,lentille-boule-seule.jpg,syst-physique-syst-optique.gif'
+---
+
+! *TU DESAFÍO*  :  Un objeto (un cuadro), un sistema físico (una lente de bola), ¿cuántos escenarios y sistemas ópticos?
+!
+! ![](syst-physique-syst-optique.gif)
+!
+! *Tiempo de descubrimiento: 30 minutos* <br>
+! *Tiempo de resolución: 10 minutos*
+! <details markdown=1><!--1-->
+!<summary>
+! Elijo esto.
+!
+! </summary>
+! Una lente bola es un sistema físico simple: una esfera de vidrio de índice de refracción $n = 1.5$ y de radio $R=5\;cm$.
+!
+! Se coloca una lente bola frente a un cuadro. Dependiendo de la posición del observador o de la cámara, el sistema óptico (la secuencia de elementos ópticos simples cruzados por la luz entre el objeto físico y la imagen observada) que forma la imagen difiere.
+!
+! Observe las 3 imágenes del cuadro por la lente bola :
+!
+! Imagen 1
+! ![](lentille-boule-transparence.jpg)
+! Imágenes 2 (la más pequeña) y 3
+! ![](lentille-boule-2-reflexions.jpg)
+!
+! Para cada una de las imágenes de la tabla, ¿puede identificar el sistema óptico y luego especificar las características de los diversos elementos simples que conforman el sistema y sus distancias relativas?
+!
+! * _El tiempo de resolución es el tiempo esperado típico que se asignará a este problema si fuera parte de un examen para un certificado óptico._
+! * _El tiempo de descubrimiento es el tiempo esperado requerido para preparar este desafío si no tienes práctica. Pero toma todo el tiempo que necesites_.
+!
+! <details markdown=1><!--2-->
+! <summary>
+! ¿Listo para responder las preguntas del equipo M3P2 para la imagen 1?
+! </summary>
+! <details markdown=1>
+! <summary>
+! ¿Dónde se encuentra el cuadro?
+! </summary>
+! * La imagen se encuentra en el otro lado de la lente bola, en relación con usted.
+!</details>
+!
+! <details markdown=1>
+! <summary>
+! ¿Cuál es el sistema óptico que da la imagen 1 del cuadro?
+! </summary>
+! * El sistema óptico consta de dos superficies de refracción esféricas centradas en el mismo eje.
+!</details>
+!
+! <details markdown=1>
+! <summary>
+! ¿Cómo caracterizas cada uno de los elementos ópticos simples que componen el sistema óptico, y la distancia relativa entre ellos?
+! </summary>
+! * El eje óptico está orientado positivamente en la dirección de propagación de la luz (desde el cuadro hasta la lente de bola).
+!
+! * La primera superficie de refracción esferica $DS1$ encontrada por la luz tiene las siguientes características :<br>
+! $\overline{S_1C_1}=+|R|=+5\;cm$ , $n_{ini}=1$ and $n_{fin}=1.5$
+!
+! * La segunda superficie de refracción esferica $DS2$ encontrada por la luz tiene las siguientes características :<br>
+! $\overline{S_2C_2}=-|R|=-5\;cm$ , $n_{ini}=1.5$ and $n_{fin}=1$
+!
+! * La distancia algebraica entre $DS1$ y $DS2$ es : $\overline{S_1S_2}=+10\;cm$
+!</details>
+! <details markdown=1>
+! <summary>
+! Si tuvieras que determinar las características de la imagen (tamaño y posición), ¿cómo manejaría el problema?
+! </summary>
+! * $DS1$ da una imagen $B_1$ de un objecto $B$. Esta imagen $B_1$ por $DS1$ se convierte en objecto por $DS2$. $DS2$ da una imagen $B'1$ del objecto $B_1$.
+! </details>
+! </details>
+! <!--FOR IMAGES 2 & 3-->
+!
+! <details markdown=1>
+! <summary>
+!
+! ¿Listo para responder las preguntas del equipo M3P2 para las imágenes 2 y 3 del cuadro?
+! </summary>
+!
+! <details markdown=1>
+! <summary>
+! ¿Dónde se encuentra el cuadro?
+! </summary>
+! * El cuadro está ubicado en el mismo lado de la lente de bola que tú, detrás de ti.
+! </details>
+! <details markdown=1>
+! <summary>
+! ¿Cuáles son los dos sistemas ópticos en el origen de estas dos imágenes? ¿Y puedes caracterizar cada uno de los elementos ópticos simples (+ sus distancias relativas) que constituyen cada uno de estos dos sistemas ópticos?
+! </summary>
+! * El primer sistema óptico $SO1$ consiste en un espejo esferico convexo simple (el objeto se refleja en la parte frontal de la lente de bola). Al mantener el eje óptico orientado positivamente en la dirección de propagación de la luz incidente sobre la lente de bola, el valor algebraico del radio del espejo es : $\overline{SC}=+5\;cm$.
+! * El segundo sistema óptico $SO2$ está compuesto por tres elementos ópticos simples :<br><br>
+! 1) La luz pasa primero a través de la superficie refractiva esférica $DS1$ de características : $\overline{S_1C_1}=+|R|=+5\;cm$ , $n_{ini}=1$ and $n_{fin}=1.5$.<br><br>
+! 2) La luz se refleja dentro de la segunda superficie de la lente, que actúa como un espejo esférico cóncavo de características. : $\overline{S_2C_2}=-|R|=-5\;cm$, $n=1.5$.<br><br>
+! 3) Finalmente, la luz cruza nuevamente (en la dirección opuesta) la primera superficie esferica de la lente bola que actúa como una superficie refractiva esférica de características : $\overline{S_3C_3}=+|R|=+5\;cm$ , $n_{ini}=1.5$ and $n_{fin}=1$.<br><br>
+! Las distancias algebraicas entre los diferentes elementos de $SO2$ son :<br>
+! $\overline{S_1S_2}=+10\;cm$ and $\overline{S_2S_3}=-10\;cm$
+! </details>
+! <details markdown=1>
+! <summary>
+! ¿Qué imagen está asociada con cada uno de estos dos sistemas ópticos?
+! </summary>
+! * Es difícil estar 100% convencido antes de hacer los cálculos.
+! </details>
+! <details markdown=1>
+! <summary>
+! ¿Por qué tomaste esta última foto en la oscuridad, con solo el cuadro iluminado detrás de la cámara, para obtener las imágenes 2 y 3?
+! </summary>
+! * En una superficie refractiva, parte del poder de la luz incidente se refracta y la otra se refleja. Para un material transparente y en el rango visible, la porción de potencia reflejada sigue siendo pequeña. Si la habitación se hubiera iluminado de manera homogénea como la luz del día, las imágenes 2 y 3 del cuadro detrás de la cámara habrían sido difíciles de ver frente a la imagen de la pared frontal a través de la lente bola.
+! </details>
+! </details>
+! </details>
+
+
+
+
+!!!! *PUNTO DIFÍCIL* (contribuir o indicar los puntos difíciles que merecen explicarse mejor)
+!!!!
+
+!! MÀS ALLÁ* (contribuir)
+!!
+
+!! *ESCAPE CULTURAL* (contribuir)
+!!
+
+!!! *MAÎTRISES-TU ?*
+!!! <details markdown = 1>
+!!! <summary>
+!!! Descrire le test
+!!! </summary>
+!!! Texte du test (et ses images, figures, audio, video, etc ...)
+!!! ! [a figure for the test] (file-name)
+!!!
+!!! Texte de la question
+!!! <details markdown=1>
+!!! <summary>
+!!! Choix de réponse 1
+!!! </summary>
+!!! Texte si la réponse 1 est choisie
+!!! </details>
+!!! <details markdown=1>
+!!! <summary>
+!!! Choix de réponse 2
+!!! </summary>
+!!! Texte si la réponse 2 est choisie
+!!! </details>
+!!! <!--possibility to add answers, questions-->
+!!! -----
+!!! <details markdown=1>
+!!! <summary>
+!!! Solution complète
+!!! </summary>
+!!! Texte de la solution
+!!! </details>
diff --git a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/03.physical-system-versus-optical-system/03.new-course-beyond/annex.fr.md b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/03.physical-system-versus-optical-system/03.new-course-beyond/annex.fr.md
new file mode 100644
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--- /dev/null
+++ b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/03.physical-system-versus-optical-system/03.new-course-beyond/annex.fr.md
@@ -0,0 +1,154 @@
+---
+title: 'new course : beyond'
+media_order: 'syst-physique-syst-optique.gif,lentille-boule-seule.jpg,lentille-boule-transparence.jpg,lentille-boule-2-reflexions.jpg'
+---
+
+! *TON  DÉFI*  :  Un objet (un tableau), un système physique (une lentille boule), et combien de scénarios et systèmes optiques?
+!
+! ![](syst-physique-syst-optique.gif)
+!
+! *Temps de découverte : 30 minutes*<br>
+! *Temps de résolution : 10 minutes*
+! <details markdown=1><!--1-->
+!<summary>
+! Je choisis.
+!
+! </summary>
+! Une lentille boule est un système physique simple : une sphère de verre d'indice de réfraction $n=1.5$ et de rayon $R=5\;cm$.
+!
+! La lentille boule est placée devant un tableau. Selon la position de l'observateur ou de l'objectif d'un appareil photo, le système optique (l'enchaînement des éléments optiques simples traversés par la lumière entre l'objet physique et l'image observée) qui forme l'image diffère.
+!
+! Observe les 3 images du tableau données par la lentille boule :
+!
+! Image 1
+! ![](lentille-boule-transparence.jpg)
+! Images 2 (la plus petite) et 3
+! ![](lentille-boule-2-reflexions.jpg)
+!
+! Pour chacune des images du tableau, saurais-tu identifier le système optique, puis préciser les caractéristiques des différents éléments simples qui composent le système et leurs distances relatives?
+!
+! * _Le temps de résolution est le temps typique qui serait alloué à ce genre de problème dans un examen d'optique_. 
+! * _Le temps de découverte est un temps estimé pour préparer ce défi si tu découvres l'optique. Il ne s'agit que d'un temps indicatif, prends autant de temps que nécesaire_.
+!
+!<\details>
+! <details markdown=1>
+! <summary>
+! Prêt à répondre aux questions de l'équipe M3P2 concernant l'image 1?
+! </summary>
+!
+! <details markdown=1>
+! <summary>
+! Où est localisé le tableau ?
+! </summary>
+! * L'image est située de l'autre côté de la lentille boule, par rapport à toi.
+!</details>
+!
+! <details markdown=1>
+! <summary>
+! Quel est le système optique qui donne l'image 1 du tableau ?
+! </summary>
+! * Le système optique est constitué de deux dioptres sphériques centrés sur un même axe.
+!</details>
+!
+! <details markdown=1>
+! <summary>
+! Comment caractérises-tu chacun des éléments optiques simples qui constituent le système optique, et la distance relative entre eux? 
+! </summary>
+! * L'axe optique est orienté positivement dans le sens de propagation de la lumière (du tableau vers la lentille boule).
+!
+! * Le premier dioptre sphérique $DS1$ rencontré par la lumière possède les caractéristiques suivantes :<br>
+! $\overline{S_1C_1}=+|R|=+5\;cm$ , $n_{ini}=1$ and $n_{fin}=1.5$
+!
+! * Le deuxième dioptre sphérique $DS2$ erencontré par la lumière possède les caractéristiques suivantes :<br>
+! $\overline{S_2C_2}=-|R|=-5\;cm$ , $n_{ini}=1.5$ and $n_{fin}=1$
+!
+! * La distance algébrique entre $DS1$ et $DS2$ est : $\overline{S_1S_2}=+10\;cm$
+!</details>
+! <details markdown=1>
+! <summary>
+! Si tu devais déterminer les caractéristiques de l'image (taille et position), comment prendrais-tu le problème?
+! </summary>
+! * $DS1$ donne une image $B_1$ d'un objet $B$. Cette image $B_1$ pour $DS1$ devient object pour $DS2$. $DS2$ donne une image $B'1$ de l'objet $B_1$.
+! </details>
+! </details>
+! <!--FOR IMAGES 2 & 3-->
+!
+! <details markdown=1>
+! <summary>
+!
+! Prêt à répondre aux questions de l'équipe M3P2 pour les images 2 et 3 du tableau?
+! </summary>
+!
+! <details markdown=1>
+! <summary>
+! Où le tableau est-il localisé?
+! </summary>
+! * Le tableau est localisé du même côté de la lentille boule que toi, et derrière toi.
+! </details>
+! <details markdown=1>
+! <summary>
+! Quels sont les deux systèmes optiques à l'origine de ces deux images? Et peux tu caractériser chacun des éléments optiques simples (+ leurs distances relatives) qui constituent chacun de ces deux systèmes optiques ?
+! </summary>
+! * Un premier système optique $OS1$ se compose d'un simple miroir convexe (l'objet se reflète sur la face avant de la lentille boule). En gardant l'acxe optique orienté positivement dans le sens de propagation de la lumière incidente sur la lentille boule, la valeur algébrique du rayon du miroir est : $\overline{SC}=+5\;cm$.
+! * Le second système optique $OS2$ est composé de trois éléments optiques simples :<br><br>
+! 1) La lumière travrse en premier le dioptre sphérique $DS1$ de caractéristiques : $\overline{S_1C_1}=+|R|=+5\;cm$ , $n_{ini}=1$ and $n_{fin}=1.5$.<br><br>
+! 2) La lumière est alors réfléchie à l'intérieur sur la deuxième surface de la lentille, qui agit comme un mirroir sphérique de caractéristiques : $\overline{S_2C_2}=-|R|=-5\;cm$, $n=1.5$.<br><br>
+! 3) Finalement la lumière retraverse (en ens inverse) la première surface de la lentille boule qui agit comme un dioptre sphérique de caractéristiques : $\overline{S_3C_3}=+|R|=+5\;cm$ , $n_{ini}=1.5$ and $n_{fin}=1$.<br><br>
+! les distances algébriques entre les différents éléments de $OS2$ sont :<br>
+! $\overline{S_1S_2}=+10\;cm$ and $\overline{S_2S_3}=-10\;cm$
+! </details>
+! <details markdown=1>
+! <summary>
+! Quelle image est associée à chacun de ces deux systèmes optiques?
+! </summary>
+! * Il est difficile de se convaincre à 100% avant d'avoir fait les calculs. 
+! </details>
+! <details markdown=1>
+! <summary>
+! Pourquoi avoir pris cette dernière photo dans le noir, avec seulement derrière l'appareil photo le tableau illuminé, pour obtenir les images 2 et 3?
+! </summary>
+! * Sur un dioptre, une partie de la puissance de la lumière incidente est réfractée, et l'autre est réfléchie. Pour un matérieu transparent et dans le domaine visible, la part de puissance réfléchie reste petite. Si la pièce avait été éclairée de manière homogène comme en plein jour, les images 2 et 3 du tableau situé derrière l'appareil photo auraient été difficilement visibles devant l'image du mur de devant à travers la lentille boule. 
+! </details>
+! </details>
+! </details>
+
+
+
+!!!! *DIFFICULT POINT* (contribute, or indicate a difficult point of understanding)
+!!!!
+
+!! *BEYOND* (to contribute)
+!!
+
+!! *CULTURAL POINT* (contributor)
+!!
+
+!!! *MAÎTRISES-TU ?*
+!!! <details markdown = 1>
+!!! <summary>
+!!! Descrire le test
+!!! </summary>
+!!! Texte du test (et ses images, figures, audio, video, etc ...)
+!!! ! [a figure for the test] (file-name)
+!!!
+!!! Texte de la question
+!!! <details markdown=1>
+!!! <summary>
+!!! Choix de réponse 1
+!!! </summary>
+!!! Texte si la réponse 1 est choisie
+!!! </details>
+!!! <details markdown=1>
+!!! <summary>
+!!! Choix de réponse 2
+!!! </summary>
+!!! Texte si la réponse 2 est choisie
+!!! </details>
+!!! <!--possibility to add answers, questions-->
+!!! -----
+!!! <details markdown=1>
+!!! <summary>
+!!! Solution complète
+!!! </summary>
+!!! Texte de la solution
+!!! </details>
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diff --git a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/03.physical-system-versus-optical-system/topic.en.md b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/03.physical-system-versus-optical-system/topic.en.md
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@@ -0,0 +1,4 @@
+---
+title: 'Physical systems versus optical systems '
+---
+
diff --git a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/03.physical-system-versus-optical-system/topic.es.md b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/03.physical-system-versus-optical-system/topic.es.md
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@@ -0,0 +1,5 @@
+---
+title: 'Sistema físico versus sistema óptico'
+media_order: aef-optics-test4.mov
+---
+
diff --git a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/03.physical-system-versus-optical-system/topic.fr.md b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/03.physical-system-versus-optical-system/topic.fr.md
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@@ -0,0 +1,4 @@
+---
+title: 'Système physique versus sytème optique'
+---
+
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new file mode 100644
index 000000000..1a131d64c
--- /dev/null
+++ b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/01.new-course-main/textbook.es.md
@@ -0,0 +1,5 @@
+---
+title: 'nuevo curso: principal'
+---
+
+nuevo curso: principal
\ No newline at end of file
diff --git a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/01.new-course-main/textbook.fr.md b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/01.new-course-main/textbook.fr.md
new file mode 100644
index 000000000..305ebaeeb
--- /dev/null
+++ b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/01.new-course-main/textbook.fr.md
@@ -0,0 +1,5 @@
+---
+title: 'nouveau cours : principal'
+---
+
+nouveau cours : principal
\ No newline at end of file
diff --git a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/02.new-course-overview/axe_opt.gif b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/02.new-course-overview/axe_opt.gif
new file mode 100644
index 000000000..fcc23d8b6
Binary files /dev/null and b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/02.new-course-overview/axe_opt.gif differ
diff --git a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/03.paraxial-optics-centered-systems/02.new-course-overview/cheatsheet.en.md b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/02.new-course-overview/cheatsheet.en.md
similarity index 100%
rename from 01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/03.paraxial-optics-centered-systems/02.new-course-overview/cheatsheet.en.md
rename to 01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/02.new-course-overview/cheatsheet.en.md
diff --git a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/03.paraxial-optics-centered-systems/02.new-course-overview/cheatsheet.es.md b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/02.new-course-overview/cheatsheet.es.md
similarity index 100%
rename from 01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/03.paraxial-optics-centered-systems/02.new-course-overview/cheatsheet.es.md
rename to 01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/02.new-course-overview/cheatsheet.es.md
diff --git a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/03.paraxial-optics-centered-systems/02.new-course-overview/cheatsheet.fr.md b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/02.new-course-overview/cheatsheet.fr.md
similarity index 100%
rename from 01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/03.paraxial-optics-centered-systems/02.new-course-overview/cheatsheet.fr.md
rename to 01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/02.new-course-overview/cheatsheet.fr.md
diff --git a/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/01.mirror/03.new-course-beyond/annex.en.md b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/03.new-course-beyond/annex.en.md
similarity index 100%
rename from 01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/01.mirror/03.new-course-beyond/annex.en.md
rename to 01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/03.new-course-beyond/annex.en.md
diff --git a/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/01.mirror/03.new-course-beyond/annex.es.md b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/03.new-course-beyond/annex.es.md
similarity index 100%
rename from 01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/01.mirror/03.new-course-beyond/annex.es.md
rename to 01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/03.new-course-beyond/annex.es.md
diff --git a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/03.paraxial-optics-centered-systems/03.new-course-beyond/annex.fr.md b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/03.new-course-beyond/annex.fr.md
similarity index 100%
rename from 01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/03.paraxial-optics-centered-systems/03.new-course-beyond/annex.fr.md
rename to 01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/03.new-course-beyond/annex.fr.md
diff --git a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/04.new-course-parallel-1/default.en.md b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/04.new-course-parallel-1/default.en.md
new file mode 100644
index 000000000..c5ea2af52
--- /dev/null
+++ b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/04.new-course-parallel-1/default.en.md
@@ -0,0 +1,52 @@
+---
+title: 'new course : parallel 1'
+content:
+    items: '- ''@self.children'''
+    order:
+        by: date
+        dir: desc
+    limit: '5'
+    pagination: '1'
+    url_taxonomy_filters: '1'
+hero_classes: ''
+hero_image: ''
+---
+
+new course : parallel 1
+
+#####for parallel or other level course :
+It will be possible to redirect towards an other page in the m3p2 cursus.
+Or to write here some paragraphes separated by html from an other pages, I think.
+Or write here a completely new parallel course
+
+For the moment, please wait.
+
+Ya es possible hacer
+I can here :
+- write a full "parallel 1 course" if required
+- or add add a few word about a "course parallel 1" that is written somewhere in "pages/curriculum/..."
+- or do nothing.
+
+    Go 
+    
+    [there](http://localhost:8000/en/m3p2-curriculum/physics-chemistry-biology/niv3/Geometrical-optics/geometrical-optics-general/geometrical-optics-validity/geometrical-optics-domain-of-validity-overview)
+    
+    [Current chapter](.)
+    [Parent chapter](..)
+    [Sibling chapter](../another-chapter)
+    [Child chapter](chapter)
+    [Anchor in the page](#slug-of-header)
+
+<details markdown=1>
+<summary>
+TOWARDS parallel 1, if their are several
+</summary>
+This is a text
+```math
+f\colon\left\{\begin{aligned}\mathbb{R}_4[X]&\longrightarrow\mathbb{R}_4[X] \\
+P&\longmapsto P’\end{aligned}\right.
+\qquad
+g\colon\left\{\begin{aligned}\mathbb{R}_2[X]&\longrightarrow\mathbb{R}_2[X] \\
+P&\longmapsto XP’+P\end{aligned}\right.
+```
+</details>
diff --git a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/04.new-course-parallel-1/default.es.md b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/04.new-course-parallel-1/default.es.md
new file mode 100644
index 000000000..c10bf6246
--- /dev/null
+++ b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/04.new-course-parallel-1/default.es.md
@@ -0,0 +1,23 @@
+---
+title: 'nuevo curso : paralelo 1'
+hero_classes: ''
+hero_image: ''
+content:
+    items: '- ''@self.children'''
+    limit: '5'
+    order:
+        by: date
+        dir: desc
+    pagination: '1'
+    url_taxonomy_filters: '1'
+---
+
+nuevo curso : paralelo 1
+
+
+##### para un curso paralelo o de otro nivel:
+Será (ya es) posible redirigir hacia otra página en el cursus m3p2.
+O escribir aquí algunos párrafos separados por html de otras páginas, creo.
+Tambien podemos aquí escribir un curso paralelo o de otro nivel completamente nuevo
+
+*Por el momento, por favor espere*
\ No newline at end of file
diff --git a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/04.new-course-parallel-1/default.fr.md b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/04.new-course-parallel-1/default.fr.md
new file mode 100644
index 000000000..5f25d7fed
--- /dev/null
+++ b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/04.new-course-parallel-1/default.fr.md
@@ -0,0 +1,22 @@
+---
+title: 'nouveau cours : parallèle 1'
+hero_classes: ''
+hero_image: ''
+content:
+    items: '- ''@self.children'''
+    limit: '5'
+    order:
+        by: date
+        dir: desc
+    pagination: '1'
+    url_taxonomy_filters: '1'
+---
+
+nouveau cours : parallèle 1
+
+##### pour un cours parallèle ou d'un autre niveau
+Il sera posible (cela l'est déjà) de rediriger automatiquement vers une autre page dans le cursus m3p2.
+Ou d'écrire quelques paragraphes avec entre du html emprunter à d'autres pages je pense.
+On peut toujours aussi écrire ici un cours parallèle ou d'un autre niveau complétement nouveau
+
+*Pour le moment,, attendre, SVP*
\ No newline at end of file
diff --git a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/05.new-course-parallel-2/form.en.md b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/05.new-course-parallel-2/form.en.md
new file mode 100644
index 000000000..d717c9a0a
--- /dev/null
+++ b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/05.new-course-parallel-2/form.en.md
@@ -0,0 +1,12 @@
+---
+title: 'new course : parallel 2'
+---
+
+new course : parallel 2
+
+#####for parallel or other level course :
+It will be possible to redirect towards an other page in the m3p2 cursus.
+Or to write here some paragraphes separated by html from an other pages, I think.
+Or write here a completely new parallel course
+
+For the moment, please wait.
\ No newline at end of file
diff --git a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/05.new-course-parallel-2/form.es.md b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/05.new-course-parallel-2/form.es.md
new file mode 100644
index 000000000..e53f6f78b
--- /dev/null
+++ b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/05.new-course-parallel-2/form.es.md
@@ -0,0 +1,12 @@
+---
+title: 'nuevo curso : paralelo 2'
+---
+
+nuevo curso : paralelo 2
+
+##### para un curso paralelo o de otro nivel:
+Será (ya es) posible redirigir hacia otra página en el cursus m3p2.
+O escribir aquí algunos párrafos separados por html de otras páginas, creo.
+Tambien podemos aquí escribir un curso paralelo o de otro nivel completamente nuevo
+
+*Por el momento, por favor espere*
\ No newline at end of file
diff --git a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/05.new-course-parallel-2/form.fr.md b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/05.new-course-parallel-2/form.fr.md
new file mode 100644
index 000000000..cd5868bd3
--- /dev/null
+++ b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/05.new-course-parallel-2/form.fr.md
@@ -0,0 +1,12 @@
+---
+title: 'nouveau cours : parallèle 2'
+---
+
+nouveau cours : parallèle 2
+
+##### pour un cours parallèle ou d'un autre niveau
+Il sera posible (cela l'est déjà) de rediriger automatiquement vers une autre page dans le cursus m3p2.
+Ou d'écrire quelques paragraphes avec entre du html emprunter à d'autres pages je pense.
+On peut toujours aussi écrire ici un cours parallèle ou d'un autre niveau complétement nouveau
+
+*Pour le moment,, attendre, SVP*
\ No newline at end of file
diff --git a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/06.course-level-down/page.en.md b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/06.course-level-down/page.en.md
new file mode 100644
index 000000000..70107fd31
--- /dev/null
+++ b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/06.course-level-down/page.en.md
@@ -0,0 +1,14 @@
+---
+title: 'course : level-1'
+---
+
+course : level-1 
+- or level+1 if current level = 1
+- or level-2  if current level = 4
+
+#####for parallel or other level course :
+It will be possible to redirect towards an other page in the m3p2 cursus.
+Or to write here some paragraphes separated by html from an other pages, I think.
+Or write here a completely new parallel course
+
+For the moment, please wait.
\ No newline at end of file
diff --git a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/06.course-level-down/page.es.md b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/06.course-level-down/page.es.md
new file mode 100644
index 000000000..275c981f4
--- /dev/null
+++ b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/06.course-level-down/page.es.md
@@ -0,0 +1,14 @@
+---
+title: 'curso : nivel-1'
+---
+
+curso : nivel-1
+- or level+1 if current level = 1
+- or level-2  if current level = 4
+
+##### para un curso paralelo o de otro nivel:
+Será (ya es) posible redirigir hacia otra página en el cursus m3p2.
+O escribir aquí algunos párrafos separados por html de otras páginas, creo.
+Tambien podemos aquí escribir un curso paralelo o de otro nivel completamente nuevo
+
+*Por el momento, por favor espere*
\ No newline at end of file
diff --git a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/06.course-level-down/page.fr.md b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/06.course-level-down/page.fr.md
new file mode 100644
index 000000000..f3b1ebb60
--- /dev/null
+++ b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/06.course-level-down/page.fr.md
@@ -0,0 +1,14 @@
+---
+title: 'cours : niveau-1'
+---
+
+cours : niveau-1
+- or level+1 if current level = 1
+- or level-2  if current level = 4
+
+##### pour un cours parallèle ou d'un autre niveau
+Il sera posible (cela l'est déjà) de rediriger automatiquement vers une autre page dans le cursus m3p2.
+Ou d'écrire quelques paragraphes avec entre du html emprunter à d'autres pages je pense.
+On peut toujours aussi écrire ici un cours parallèle ou d'un autre niveau complétement nouveau
+
+*Pour le moment,, attendre, SVP*
\ No newline at end of file
diff --git a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/07.course-level-up/portal.en.md b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/07.course-level-up/portal.en.md
new file mode 100644
index 000000000..4f2a994ae
--- /dev/null
+++ b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/07.course-level-up/portal.en.md
@@ -0,0 +1,18 @@
+---
+title: 'course : level+1'
+hero_classes: ''
+hero_image: ''
+---
+
+course : level+1
+
+or level+2 if current level=1
+
+or level-1 if current level=4
+
+#####for parallel or other level course :
+It will be possible to redirect towards an other page in the m3p2 cursus.
+Or to write here some paragraphes separated by html from an other pages, I think.
+Or write here a completely new parallel course
+
+For the moment, please wait.
\ No newline at end of file
diff --git a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/07.course-level-up/portal.es.md b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/07.course-level-up/portal.es.md
new file mode 100644
index 000000000..380d15135
--- /dev/null
+++ b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/07.course-level-up/portal.es.md
@@ -0,0 +1,14 @@
+---
+title: 'curso : nivel+1'
+---
+
+curso : nivel+1
+or level+2 if current level=1
+or level-1 if current level=4
+
+##### para un curso paralelo o de otro nivel:
+Será (ya es) posible redirigir hacia otra página en el cursus m3p2.
+O escribir aquí algunos párrafos separados por html de otras páginas, creo.
+Tambien podemos aquí escribir un curso paralelo o de otro nivel completamente nuevo
+
+*Por el momento, por favor espere*
\ No newline at end of file
diff --git a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/07.course-level-up/portal.fr.md b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/07.course-level-up/portal.fr.md
new file mode 100644
index 000000000..e3b3ba33e
--- /dev/null
+++ b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/07.course-level-up/portal.fr.md
@@ -0,0 +1,14 @@
+---
+title: 'cours : niveau+1'
+---
+
+cours : niveau+1
+or level+2 if current level=1
+or level-1 if current level=4
+
+##### pour un cours parallèle ou d'un autre niveau
+Il sera posible (cela l'est déjà) de rediriger automatiquement vers une autre page dans le cursus m3p2.
+Ou d'écrire quelques paragraphes avec entre du html emprunter à d'autres pages je pense.
+On peut toujours aussi écrire ici un cours parallèle ou d'un autre niveau complétement nouveau
+
+*Pour le moment,, attendre, SVP*
\ No newline at end of file
diff --git a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.physical-system-versus-optical-system/aef-optics-test4.mov b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/aef-optics-test4.mov
similarity index 100%
rename from 01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.physical-system-versus-optical-system/aef-optics-test4.mov
rename to 01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/aef-optics-test4.mov
diff --git a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/03.paraxial-optics-centered-systems/topic.en.md b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/topic.en.md
similarity index 100%
rename from 01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/03.paraxial-optics-centered-systems/topic.en.md
rename to 01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/topic.en.md
diff --git a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/03.paraxial-optics-centered-systems/topic.es.md b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/topic.es.md
similarity index 100%
rename from 01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/03.paraxial-optics-centered-systems/topic.es.md
rename to 01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/topic.es.md
diff --git a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/03.paraxial-optics-centered-systems/topic.fr.md b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/topic.fr.md
similarity index 100%
rename from 01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/03.paraxial-optics-centered-systems/topic.fr.md
rename to 01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.paraxial-optics-centered-systems/topic.fr.md
diff --git a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.physical-system-versus-optical-system/.DS_Store b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.physical-system-versus-optical-system/.DS_Store
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diff --git a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.physical-system-versus-optical-system/03.new-course-beyond/annex.en.md b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.physical-system-versus-optical-system/03.new-course-beyond/annex.en.md
deleted file mode 100644
index 6ed0fb68e..000000000
--- a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.physical-system-versus-optical-system/03.new-course-beyond/annex.en.md
+++ /dev/null
@@ -1,59 +0,0 @@
----
-title: 'new course : beyond'
----
-
-! *YOUR  CHALLENGE*  :  look at the picture, and think of the right answers to the following questions
-!
-! _Do not look at the answer, take time to think, a few days if necessary. The time to build your mental representation of the phenomenon, to formulate it in words is important, a thousand times more important than the ephemeral instant where you read the fews words of the solution._
-!
-! <details markdown=1>
-! <summary>
-! INDICE "key word"
-! </summary>
-! diffusion
-! </details>
-! <details markdown=1>
-! <summary>
-! ANSWER
-! </summary>
-! diffusion
-! </details>
-
-
-!!!! *DIFFICULT POINT* (contribute, or indicate a difficult point of understanding)
-!!!!
-
-!! *BEYOND* (to contribute)
-!!
-
-!! *CULTURAL POINT* (contributor)
-!!
-
-!!! *DO  YOU  MASTER ?*
-!!! <details markdown=1>
-!!! <Summary>
-!!! Describe the test
-!!! </summary>
-!!! The text of the test (and its images, figures, audio, video, etc ...)
-!!!
-!!! Question text
-!!! <details markdown=1>
-!!! <summary>
-!!! Answer choice 1
-!!! </summary>
-!!! Text if this answer 1 is chosen
-!!! </details>
-!!! <details markdown=1>
-!!! <summary>
-!!! Answer choice 2
-!!! </summary>
-!!! Text if this answer 2 is chosen
-!!! </details>
-!!! <!--possibility to add answers, questions-->
-!!! -----
-!!! <details markdown=1>
-!!! <summary>
-!!! Complete solution.
-!!! </summary>
-!!! text of the solution
-!!! </details>
diff --git a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.physical-system-versus-optical-system/topic.en.md b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.physical-system-versus-optical-system/topic.en.md
deleted file mode 100644
index 7a0fbe4c1..000000000
--- a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.physical-system-versus-optical-system/topic.en.md
+++ /dev/null
@@ -1,22 +0,0 @@
----
-title: 'Conditions and implications of paraxial optics'
----
-
-réénoncer des conditions. Déjà fait dans chapitre précédent ?
-Implication en terme d'approximations mathématiques : 
-Dans la limite des angles petits, alors 
-$i\simeq\sin(i)\simeq\tan(i)$ 
-où $i$ est la valeur de l'angle exprimée en radian.
-et
-$cos(i)\simeq1$
-refaire cela bien ..
-Les systèmes quasi-stignatiques deviennent stigmatiques : donc 
-à un point objet situé sur l'axe optique correspond un point image situé sur l'axe optique
-pour tout point objet situés dans un même plan perpendiculaire à l'axe optique, tous les points images correspondants sont situés dans un même plan perpendiculaire à l'axe optique
-continuité,
-image objet étendu caractérisée par sa position, son grandissement transversale et son sens, grandissement longitudinal,
-formule de conjugaison donne la position
-grandissement transversale donne sa taille transverse
-grandissement longitudinal son élongation dans le sens de l'axe optique,
-... tout ca
-
diff --git a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.physical-system-versus-optical-system/topic.es.md b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.physical-system-versus-optical-system/topic.es.md
deleted file mode 100644
index 46da034a8..000000000
--- a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.physical-system-versus-optical-system/topic.es.md
+++ /dev/null
@@ -1,25 +0,0 @@
----
-title: 'Les conditions et implications de l''optique paraxiale'
-media_order: aef-optics-test4.mov
----
-
-réénoncer des conditions. Déjà fait dans chapitre précédent ?
-![](aef-optics-test4.mov)
-Implication en terme d'approximations mathématiques : 
-Dans la limite des angles petits, alors 
-$i\simeq\sin(i)\simeq\tan(i)$ 
-où $i$ est la valeur de l'angle exprimée en radian.
-et
-
-$cos(i)\simeq1$
-refaire cela bien ..
-Les systèmes quasi-stignatiques deviennent stigmatiques : donc 
-à un point objet situé sur l'axe optique correspond un point image situé sur l'axe optique
-pour tout point objet situés dans un même plan perpendiculaire à l'axe optique, tous les points images correspondants sont situés dans un même plan perpendiculaire à l'axe optique
-continuité,
-image objet étendu caractérisée par sa position, son grandissement transversale et son sens, grandissement longitudinal,
-formule de conjugaison donne la position
-grandissement transversale donne sa taille transverse
-grandissement longitudinal son élongation dans le sens de l'axe optique,
-... tout ca
-
diff --git a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.physical-system-versus-optical-system/topic.fr.md b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.physical-system-versus-optical-system/topic.fr.md
deleted file mode 100644
index ff6d64867..000000000
--- a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/05.paraxial-optics/04.physical-system-versus-optical-system/topic.fr.md
+++ /dev/null
@@ -1,22 +0,0 @@
----
-title: 'Les conditions et implications de l''optique paraxiale'
----
-
-réénoncer des conditions. Déjà fait dans chapitre précédent ?
-Implication en terme d'approximations mathématiques : 
-Dans la limite des angles petits, alors 
-$i\simeq\sin(i)\simeq\tan(i)$ 
-où $i$ est la valeur de l'angle exprimée en radian.
-et
-$cos(i)\simeq1$
-refaire cela bien ..
-Les systèmes quasi-stignatiques deviennent stigmatiques : donc 
-à un point objet situé sur l'axe optique correspond un point image situé sur l'axe optique
-pour tout point objet situés dans un même plan perpendiculaire à l'axe optique, tous les points images correspondants sont situés dans un même plan perpendiculaire à l'axe optique
-continuité,
-image objet étendu caractérisée par sa position, son grandissement transversale et son sens, grandissement longitudinal,
-formule de conjugaison donne la position
-grandissement transversale donne sa taille transverse
-grandissement longitudinal son élongation dans le sens de l'axe optique,
-... tout ca
-
diff --git a/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/06.eye-correction/.DS_Store b/01.curriculum/01.physics-chemistry-biology/03.Niv3/02.Geometrical-optics/06.eye-correction/.DS_Store
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new file mode 100644
index 000000000..a24e3e6b6
--- /dev/null
+++ b/01.curriculum/01.physics-chemistry-biology/03.Niv3/topics.en.md
@@ -0,0 +1,5 @@
+---
+title: hills
+slug: hills
+---
+
diff --git a/01.curriculum/01.physics-chemistry-biology/03.Niv3/topics.fr.md b/01.curriculum/01.physics-chemistry-biology/03.Niv3/topics.fr.md
new file mode 100644
index 000000000..940548ccf
--- /dev/null
+++ b/01.curriculum/01.physics-chemistry-biology/03.Niv3/topics.fr.md
@@ -0,0 +1,5 @@
+---
+title: collines
+slug: hills
+---
+
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-title: 'Preparation skype 2019-04'
----
-Dear Claudia, Oscar, Andres, Neil
-
-For this skype we add to ourselves (Cesar, Pier Francesco and Claude) Antoine (Web developer who joined the project team) and Jean-Yves Plantec (Director of the C2IP, "Centre d'Innovation et d'Ingénierie Pédagogique" à l'INSA-Toulouse)
-
-Welcome to the (temporary) site for the project development.
-
-This lines below are to prepare our next skype.
-
-We propose three parts:
-1. News
-2. The objectives of the project, and forecast schedule for its realization
-3. The discussion points (please, fill free to add some)
-
-(the forecast schedule concerning the achievements of the first year, is based on a Franco-French budget of 25k€, intended exclusively for IT development and site hosting.)
-
-
-## OBJECTIVES OF THE PROJECT, AND FORECAST SCHEDULE
-
-### Initial objectives of the project
-
-#### "online courses, usable on computers, tablets, and with ergonomics compatible with smartphones"
-<ins>Target of achievement at the end of the first year :</ins> will be done, at least in a first "beta" version. 
-
-#### "The target audience is, in a general manner,  young people between 15 and 20 years of age and young adults in a phase of conversion"
-
-<ins>Target of achievement at the end of the first year :</ins>
-1. a "hidden" online beta version will be proposed as soon as September 2019 to all students in the partner institutes or professors in the partner institutes or the teams. At INSA we need it concerning the "Geometrical optics" theme for our specific groups IberINSA (courses in spanich), EngINSA (courses in english), Norginsa (courses in French). Of course this "Geometrical Optics" theme will evolve and be improved with time, it will take 2 or 3 years for the pedagogical team to define and built a consistent teaching in that field. Anyway, this will enables to test and validate the site and the courses, with a feedback from the students.
-2. the "visible" beta version about Geometrical optics will be online as soon as possible, at the latest September 2010, when the siteweb and the courses will be validated mature enough. 
-
-<ins>Implementation schedule for the following years :</ins> 
-1. The teams in the various themes will be able to have a "hidden" version for their use and students, and be able to put it visible online when they feel ready.
-2. the site will be completed, its ergonomics will be improved thanks to the feedback of the first users, and in agreement with all the partner institutes.
-
-#### "allowing to continue studying the courses and to do exercises without internet connection (after a first connection)"
-<ins>Target of achievement at the end of the first year :</ins>
-1. It seems possible to the web engineer of the project (Antoine), to have the possibility for the user to go on the study the courses off-line as soon as the end of this first year (to be confirm with time). However this will be limited to the courses, and not for the exercices (auto-evaluation).
-
-<ins>Implementation schedule for the following years :</ins>
-1. The possibility (for the users and for the partner institutes) to define "programs" will be implemented later : this require a minimum of available courses in the cursus. Anyway, this possibility de define "programs" is a strong requirement that interests the universities. This is the second phase of the project.
-2. The possibility for a user to do self-assessments and be guided to realize his program will be implemented in a third phase of the project.
-
-#### "A progressive teaching  in 4 levels (From a basic level equivalent to baccalaureate minus 3 years up to a higher level bac. + 2 years)."
-1. This is a specificity and a requirement for the project. All courses on a specific theme are realized by a same pedagogical team (for a maximum of coherence) on 4 levels of increasing mastering.
-2. From discussion with colleagues, the upper level must be equivalent to a end of Bachelor level. So the curriculum should cover the whole Bachelor level in a scientifical field. This will be most useful for professors and their students, and more useful for students who have to study prerequisites for an universitary exchange at master level. So it is not a higher level at "BAC +2 years" but at "BAC +3 to 5 years".
-3. The Geometrical optics course that INSA needs quickly will be done over the first 3 levels.
-4. The exact number of levels can vary "3, 4 or 5) dependening of the subject. This is not a problem in the structure of the site.
-
-#### "A teaching in 4 languages, developed by small international teams on each theme (ideal: Colombia and Mexico for Latin America, France and Norway for Europe)"
-<ins>Target of achievement at the end of the first year :</ins> The beta version of "Geometrical optics" required for September 2019 will be realized in 3 languages : Spanish, English and French. For the fourth language, see the discussion point "Opening possible for other countries?".
-
-#### "Diferentes posibilidades de exposición de los  cursos para adaptarse a las necesidades pedagogicas de cada alumno (cada curso está estructurado en 3 partes : ver "proyecto:pedagogía" en el wiki)"
- <ins>Target of achievement at the end of the first year :</ins> yes, should be available for the beta version of the site.
-
-### Additional objectives
-
-#### INSA Director’s Attention (Bertrand Raquet) : "the work undertaken must rely on technological choices and modalities, supported/validated by C2IP(\*) and OpenINSA, to ensure continuity and consistency in all digital productions of the INSA Toulouse, serving students, INSA and other partners".
-(\* : Centro de Innovación y Ingeniería Pedagógica del INSA-Toulouse).
- We believe that this request is fully justified and should apply to all the partner institutes of the project.
-1. This adaptability and compatibility guided the technical choices. Computer and web specialists from the engineering and educational innovation centres of the various partner institutes must be brought together.
-
-
-## DISCUSSION POINTS
-
-#### A - Project management
-Comments regarding the management of the project until now? 
-Requirements and suggestions ?
-
-#### B - The partner institutes
-Necessary because by essence, the project is an international and intercultural project. Necessary to present the project to foundations/ sponsors. Required to guarantee the quality of the educational content accessible on the site.
-
-##### Openness to other institutes on the starting scientific topics? on other scientific topics?
-1. French side: Interested contacts at Toulouse III University, on the side of life sciences/ biochemistry/ genetics. May interest physics also (note: INSA-Toulouse and University-Toulouse-III are among the founding members of the "Federal University of Toulouse").
-
-##### Opening possible for other countries?
-1. Latin America use mainly 3 languages (Spanish, Portuguese and Dutch): easy to open to other interested institutes , in Colombia, Mexico and other Spanish speaking countries if you wish (Peru?)
-2. We would like, european side, to open the project to partner institute(s) / pedagogical teams in an other country, and if possible a country in northern Europe, to benefit from a teaching tradition a little more pragmatic than that of southern Europe (so in terms of the use of the mathematical tool, of theorisation). The European Union has 24 official languages (plus Norway outside the EU but in the geographical Europe). Norway is still a goal, and Pier Francesco goes to a meeting unit.eu/ ndal.no in Norway in May. Anyway, Norway is a small country and the number of possible partners is limited.  Otherwise, Germany? According to each other, which country would it be desirable to star with?
-
-##### Feedback for a partner institute ? 
-Ideas : an inovative pedagogical project , international collaboration , the partner institutes validates the pedagogical contents of the site , visibility on the site as a partner funding institute , in a second phase (when this possibility is implemented on the site) the possibility to deposit on the site its own programs (for pre-requisites, for an entrance examination, academic year programs in various specialities, etc), benefit of funding obtained in the framework of the project (for tools to create digital educational resources, for missions, ...)
-
-#### C - The pedagogical teams
-
-##### How to constitute a pedagogical team on a specific theme ?
-Ideas : there will be contributors whose institutes are not (or not yet) partners inside the project (professors in others universities, researchers in agencies and lobaratories, individual contributors, ...). 
-
-##### How to convince a professor to participate ?
-Ideas : 
-1. when someone is desired in the team, but is hesitating : give him/her full access to the collaborative tool of courses creation : seeing the course that is being built can encourage him to participate (in France we have a proverb: "the appetite comes with eating"). 
-2. The collaborative tool to create the courses is very easy to use. As easy as to create a course with a word processing software like word or openoffice. It will be possible to work off-line at home. There will be much more possibilities than in a word processing software, and all pedagogical content create by the teams in the project will be usable.  
-3. the site (smartphone-compatible) will be a wonderful tool at the service of his students, and at the service of himself, whether for a course in front of the students (very good compatibility for a video projection), for a restructuring course (to answer students questions) or guide students towards pre-requisites.
-
-##### How to create good relationships and a good team spirit ?
-Ideas : That is an important point. How can we make collaboration human and culturally enriching between the members of an educational team? How to share more than efficient work and objective ? facebook groups? regular skype ? Missions when possible ?
-
-#### D - Collaborative tool to create the curriculum and the courses
-The tool under professional development (not the prototype) is under yours eyes. We do a demonstration if we can share ours screens during the skype.
-
-#### E - The site of the project : localisation and structuration
-1. For a technical point of view of the project, see information on the wiki : https://m3p2.irap.omp.eu/doku.php?id=sidebar:techniqueweb
-2. At the moment the website is localised on the developer server (Antoine). It is under development phase alpha. As soon as it is on phase beta (or at any time) it can be transferred on any server in the partner institutes. The idea : have mirror sites between partner institutes ? 
-
-#### F - Curriculum : the thematic needs
-<ins>Our needs at INSA-Toulouse are for :</ins>
-1. Geometrical optics : with a first light usable version (3languages and 3 levels) for students in September 2019. The team can be completed now or later, the content will have to evolve. This will be a test for the ability to work in pedagogical team, to create a course over 3 levels, and for the web site. The French part of the team is identified, searching for others partners.
-2. Mechanics : already well identified team French side, + Claudia. ready to start when the team is completed and ready to start to work.
-3. Electromagnetism (include Electrostatics/magnetostatics/.../ electromagntic waves): The French part of the team is identified, searching for others partners. Target (difficult, but to try) : a first light beta version for one of the INSA specific group would be required for February 2020.
-4. We have already 2 professors who have work on a pedagogical path (4 levels from "secondary school" up to "higher education") on "nano technologies". Unpublished by interested to develop and publish (has some contact in mexico)
-5. One colleague has realized a course "sensors" in 4 levels too. 
-6. It seems there is a real need for a theme "chemistry". 
-
-<ins>What are the needs in yours sides? </ins>
-
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----
-title: 'Preparation skype 2019-04'
----
-Chers Claudia, Oscar, Andres, Neil
-
-Pour ce prochain skype nous ajoutons à nous-mêmes (Cesar, Pier Francesco and Claude) Antoine Goutenoir (Développeur web qui a rejoint l'équipe projet) and Jean-Yves Plantec (Directeur de la C2IP, "Centre d'Innovation et d'Ingénierie Pédagogique" de l'INSA-Toulouse)
-
-Bienvenue sur ce site (temporaire) de développement de la version alpha du projet.
-
-Les lignes ci-dessous sont destinées à préparer notre prochain skype.
-
-Nous proposons 3 parties :
-
-1. Les nouvelles concernant le projet.
-2. Les objectifs du projet, et le planing prévisionnel de réalisation
-3. Les points ouvets à la discussion (n'hésitez pas à en ajouter si nécessaire)
-
-(le planning prévisionnel concernant les réalisations de la première année, est basé un budget franco-français de 25k€, destiné exclusivement à du développement informatique et hébergement de site.)
-
-
-## OBJECTIFS DU PROJET,  ET PLANNING PREVISIONNEL
-
-### Objectifs initiaux du projet
-
-#### "des cours en lignes, utilisables sur ordinateur, tablette, et avec une ergonomie compatible smartphone"
-<ins>Objectif de réalisation en fin de première année :</ins> sera réalisé, au moins dans une première version beta.
-
-#### "Le public visée est, de façon générale, les jeunes entre 15 et 20 ans et les jeunes adultes en phase de reconversion"
-
-<ins>Objectif de réalisation en fin de première année :</ins> 
-1. une version online "cachée" sera proposée des septembre 2019 à tous les étudiants des instituts partenaires et des professeurs dans les instituts partenaires ou les équipes.C'est une nécessité à l'INSA-Toulouse concernant le thème "optique géométrique" pour nos groupes specifiques IberINSA (cours en espagnol), EngINSA (cours en anglais) et NorgINSA (cours en français). Bien sûre ce thème "optique géométrique" évoluera et sera amélioré avec le temps, cela prendre au moins 2 ou 3 ans à l'équipe pédagogique pour définir et construire un cours consistent sur ce thème. Cependant cette première version nous permettra de tester et valider le site et le contenu des cours, avec un retour des étudiants utilisateurs. 
-2. Une version beta "visible" de ce thème "optique géométrique" sera mise en liugne dès que possible au plus tard en septmebre 2020, quand le site web et les cours seront jugés suffisamment mâtures. 
-
-<ins>Planing prévisionnel au-delà de la première année :</ins> 
-1. Les équipes sur les différentes thématiques scientifiques pourront developper une version "cachée" du site pour eux mêmes et leurs étudiants, et mettre cette version en ligne (acces gratuit) quand ils se sentiront prêts. 
-2. le site sera complété, et son ergonomie améliorée grâce aux retours des premiers utilisateurs, en accord avec tous les instituts partenaires.
-
-#### "permettant de continuer à étudier les cours et faire les exercices sans connexion internet (après une première connexion)" 
-<ins>Objectif de réalisation en fin de première année :</ins> 
-1. Il semble possible à l'ingénieur web du projet (Antoine) d'avoir la possibilité pour l'internaute de continuer à étudier des cours hors-ligne dès la fin de la première année (le temps confirmera). cependant cette possibilité ne sera valable que pour les cours, pas pour les exercices (d'auto-évaluation). 
-
-<ins>Planing prévisionnel au-delà de la première année :</ins> 
-1. La possibilité (pour l'utilisateur du site ou les institus partenaires) de définir un "programme" sera implémentée plus tard : cela requière un minimum de cours disponibles dans le cursus. Cependant cette possibilité de définir des "programmes d'étude" intéesse fortement les universités.  Cela sera développé dans une seconde phase du projet.
-2. la possibilité pour un utilisateur de s'autoévaluer et d'être guidé pour la réalisation de son projet sera implémentée dans une troisième phase. 
-
-####"Des cours progressifs sur 4 niveaux (depuis un niveau de base équivalent à BAC-3 ans jusqu'à un niveau supérieur BAC+2 ans)."
-1. C'est une spécificité et une contrainte du projet. Tous les cours dans une thématique donnée sont réalisés par une même équipe pédagogique (pour un maximum de cohérence) sur 4 niveaux de complexité croissante. 
-2. Après discussion avec des collègues, le niveau supérieur devrait couvrir le programme d'une fin de bachelor/Licence. Cela sera plus utile pour les professeurs et leurs étudiants, et pour les étudiants qui doivent étudier des prérequis en vue d'un échange universitaire de niveau master. Ainsi le niveau supérieur n'est plus un niveau "BAC+2" mais plutôt un niveau "BAC + 3 à 5".
-3. Le cours d'optique géométrique dont l'INSA a besoin rapidement sera réalisé sur les 3 premiers niveaux.
-4. Le nombre excaxt de niveaux peut varier "3, 4 ou 5" selon les thèmes. cela n'est pas un problème dans la structuration du site. 
-
-####Un enseignement dispensé en 4 langues, élaboré par de petites équipes internationales sur chaque thème (idéal : Colombie et Mexique pour l'Amérique latine, France et Norvège pour l'Europe)
-<ins>Objectif de réalisation en fin de première année :</ins> La version beta de l'optique géométrique requise pour septembre 2019 sera réalisée en 3 langues : espagnol, anglais et français. Pour une quatrième langue, voire le point de discussion "ouverture possible vers d'autres pays?" 
-
-####"Différentes possibilités d'exposition des cours pour s'adapter au mieux aux besoins pédagogiques de l'étudiant (chaque cours est structuré en 3 parties : voir "projet:pédagogie" sur le wiki)."
-<ins>Objectif de réalisation en fin de première année :</ins> oui, devrait être implémentée dans le  version beta du site.
-
-
-### Objectifs additionnels
-
-####"Point d'attention du directeur de l'INSA :  le travail engagé doit s'appuyer sur des choix technologiques et des modalités, supportés/validées par la C2IP et OpenINSA, pour assurer la continuité et la cohérence dans l'ensemble des productions numériques de l'INSA Toulouse, au service des étudiants, de l'INSA et des autres partenaires."
-Nous pensons que cette demande est entièrement justifiée, et qu'elle doit s'appliquer à tous les instituts partenaires du projet.
-1. Cette adaptabilité et compatibilité a guidé les choix techniques. Les specialistes informatique des centres d'innovations et d'ingénierie pédagogiques des differents partenaires devraient être mis en contact. 
-
-## POINTS DE DISCUSSION
-
-#### A - Gestion du projet
-Remarques sur la gestion du projet jusqu'à maintenant? 
-Oublis? Suggestions pour améliorer ?
-
-#### B - Les instituts partenaires
-Nécessaire car par essence, le projet est un projet international et interculturel. Nécessaire pour présenter le projet aux fondations / sponsors. Nécessaire pour garantir la qualité des contenus pédagogiques accessible sur le site.
-
-##### Ouverture vers d'autres instituts partenaires sur les thèmes de départ? vers d'autres thème scientifiques?
-1. Côté français : des contacts sont intéressés par le projet, plutôt du côté des sciences de la vie, de la biochimie et de la génétique. Possible intérêt en physique aussi (note: INSA-Toulouse et l'University-Toulouse-III font partie des membres fondateurs de l'Université Fédérale de Toulouse.
-
-##### Ouverture possible vers d'autres pays?
-1. L'amérique latine utilise prioncipalement 3 langues (l'espagnol, le portugais et le néerlandais) : relativement facile d'ouvrir le projet à d'autres instituts qui seraient intéressés, en Colombie, au Mexique, ou d'autres pays hispanophone si la volonté existe (Pérou?) 
-2.Nous souhaitons, côté européen, ouvrir le projet à d'autres instituts / équipes pédagogiques dans un autre pays, si possible un pays du nord de l'Europe pour bénéficier d'une tradition d'enseignement plus pragmatique que celle de l'Europe du sud (en terme de mathématisation, de théorisation) . L'union européenne a 24 langues officielles (plus la Norvège qui appartient à l'Europe géographique). la Norvège reste un objectif et pier Francesco va à une réunion unit.eu/ndal.no en Norvège en mai. Cependant la Norvège est un pays peu peuplé et le nombre d'instituts partenaires possibles est limité. Autre possibilité, l'Allemagne?  Y a t-il des intérêt à développer des liens avec ce pays pour chacun de nous?
-
-##### Intérêt pour un institut partenaire? 
-Idées : un projet pédagogiquement innovant ; collaboration international , les instituts partenaires valident les contenus pédagogiques du site , reconnaissance sur le site en tant que institut membre fondateur , et dans une seconde phase (quand cette possibilité sera implémentée sur le site) de déposer se spropres programmes d'étude (pour des prérequis, pour des tests de sélection, des programmes annuels pour ses propres étudiants dans diverses spécialités, etc) , bénéficier de fonds obtenus dans le cardre du projet (pour des outils de création pédagogique numérique, pour des missions, ...)
-
-#### C - Les équipes pédagogiques
-
-##### Comment constituer une équipe pédagogique sur un thème donné?
-Idées : 
-1. Quand quelqu'un est souhaité dans l'équipe mais qu'il hésite : possibilité de lui donner accès total à l'outil de création des cours : en voyant les cours cela peut lui donner envie de participer (en France nous avons un proverbe : "l'appétit vient en mangeant")
-2. L'outil collaboratif de création des cours est très facile à utiliser. AAussi facile que de créer son cours avec un simple traitement de texte type word ou open-office. Il y aura aussi la possibilité de travailler hors-connexion à la maison. Il y aura bien plus de possibilité qu'avec un logiciel de traitement de texte classique, et le matériel pédagogique créé par les autres membres de l'équipe sera disponible. 
-3. le site (compatible smartphone) sera un merveilleux outil au service des ses étudiants, et à son propre service, que cela soit pour un cours devant les étudiants (bonne compatibilité pour la vidéo-projection), pour un cours inversé ou pour guider ses étudiants vers des prérequis mal maîtrisés. 
-
-
-##### Comment créer de bonnes relations eb un bon esprit d'équipe ?
-Idées : C'est un point important. Comment rendre la collaboration humaine et culturellement enrichissante entre les membres d'une équipe pédagogique? Comment partager plus efficacement le travail et l'objectif? Groupes facebook ? visios régiulières ? de smissions quand c'est possible ?
-
-#### D - Outil collaboratif de création des cours
-L'outil collaboratif plus professionnel de création de cours en développement est sous vos yeux. Nous ferons une petite démonstration si nous pouvons partager notre écran sous skype. 
-
-#### E - Le site du projet : localisation et structuration
-1. For a technical point of view of the project, see information on the wiki : https://m3p2.irap.omp.eu/doku.php?id=sidebar:techniqueweb
-2. At the moment the website is localised on the developer server (Antoine). It is under development phase alpha. As soon as it is on phase beta (or at any time) it can be transferred on any server in the partner institutes. The idea : have mirror sites between partner institutes ? 
-
-#### F - Le cursus : les besoins thématiques
-<ins>Nos besoins à l'INSA-Toulouse sont :</ins>
-1. Optique géométrique : avec une première version utilisable (3langues et 3 niveaux) pour nos étudiants en septembre 2019.
-Geometrical optics : with a first light usable version (3languages and 3 levels) for students in September 2019. L'équipe peut être complétée maintenant ou plus tard, le contenu sera amené à évoluer dans les prochaines années. Cela sera un test pour un travail collaboratif en équipe pédagogique, pour créer un cours sur 3 niveaux, et pour le site web. La partie française de l'optique géométrique est identifiée. Nous cherchons des partenaires. 
-2. Mécanique : nous avons déjà indentifié une équipe française + Claudia, prête à démarrer dès que l'équipe est complétée et que l'outil de travail collaboratif est pleinement opérationnel.
-3. Electromagnetisme (qui inclut l'électrostatique, la magnétostatique, ..., les ondes électromagnétiques): la partie française de l'équipe est identifiée, en recherche d'autres partenaires. Objectif (difficile à tenir mais à essayer) :une première version beta de ce cours serait utile pour un groupe spécifique en février 2020.
-4. Nous avons déjà deux professeurs/chercheurs qui ont réalisé un chemin pédagogique (sur 4 niveaux depuis un niveau college jusqu'à un niveau supérieur) sur les nanotechnologies. Travail non publié mais intéressement à son développement et publication sur le site (à des contacts au Mexique). 
-5. Un collègue à réalisé un cours "capteurs" sur 4 niveaux. 
-6. Il semble qu'il y a un vrai besoin pour développer un thème "chimie". 
-
-<ins>Quels sont les besoins de votre côté? </ins>
-
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