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
index a0eb54d2f..f12668838 100644
--- 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
@@ -37,70 +37,81 @@ A whole chapter to write $\Longrightarrow$ nature of light (refraction, reflexio
 
 * **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*.
+* to **realize optical images of physical objects** to be *seen with the naked eye* or to be *captured by an image sensor*.
+
+* **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** : *large* (compared to $\lambda_{optical}$) *volume* of matter (liquid or solid) whose *external surface can mentally be broken 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**.
+\- *emits* or *diffuses light in all direction* outside the volume. That means in equivalent ways :  emits a *spherical wave* (_wave optics_), emits **light rays** (_ray optics_), emits *photons* (_quantum 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*.
+* **naked eyes** = **direct vision** of an object : *pencils* from all visible point sources of the object *intercept the pupil of the eye*.
 
 ![](vision-object-b.jpg)<br>
-_Direct vision : the pencils of each visible point souces intercept the iris of my eyes_
+Fig. 1. Direct vision : the pencils of each visible point souces intercept the pupil of the eye.
 
 #### 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.
+* The object is seen from a **specific angle of view**, whose principal direction named **line of sight** defines (_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**. 
+* The part of the *light that diverge from any point source* of the object and intercept the imager, after interaction *converges  in a new point* in space named **real image point**, or *diverges from a new point* in space named **virtual image point**.
 
-* **Image** : *set of all the image points*. 
+* **Image** : *set of all real and virtual 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._
+Fig. 2a. Image vision : an optical imager (rectangle) has modified the incident pencils. Only pencils from image points enter my eyes. here the image points are real.
 
+![](image_virtuelle.jpg)<br>
+Fig. 2b. here the image points are virtual.
 
 #### 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.
+* **Intercept part of the light** emitted or diffused by the physical object.
+
+* *modify the pencils* of light from each object point, to give a **new pencil** that  :<br>
+\- *converges* into a new point in space $\Longrightarrow$ *concentration of light energy* at this point.
+\- *diverges* from on other new point in space $\Longrightarrow$ *no concentration of light energy* at this point.
 
 ** Optical imager** :
-* create a **real three dimensional** *image* of the extended object surface oriented towards the imager.
-* use **refraction and/or reflexion** phenomena.
+* create a **true three dimensional** *image* of the extended object surface oriented towards the imager.
+* use **refraction and/or reflection** 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**
+* An imager can be : individual **thin simple optical element** or **centered optical system**
 
 ![](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._
+Fig. 3a. Using refraction and/or reflection phenomena, an optical imager modifies all incident pencils from each point source,to give emerging pencils that actually focus the light energy of the source points into real image points, or pencils that diverge from virtual image points. Here a real image is observed.
+
+
+![](Object-image_virtual.jpg)<br>
+Fig. 3b.  Here a virtual image is observed.
 
 #### 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_
+Fig. 4. Simple optical element : refracting or reflecting element, rotationaly symmetrical around an axis.
 
-* **Thin** $\Longrightarrow$ *diameter $\gg$ thickness or depth*)
+* **Thin** $\Longrightarrow$ *diameter $\gg$ thickness or depth*.
 
-* **Simple** : surfaces of *simple* optical element are *plane or spherical*
+* **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>
+\- *plane or thin spherical refracting surfaces*.<br>
+\- *plane or thin spherical mirrors*.<br>
 \- *thin lenses*.
 
 #### Centered optical systems
@@ -110,17 +121,17 @@ _simple optical element : refracting or reflecting element, rotationaly symmetri
 * *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_
+Fig. 5. Centered 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
+#### From mathematical idealization to physical 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>
+\- extension of the volume can not be resolved with naked eye.<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**).
@@ -129,26 +140,33 @@ has a *location* in space, but *no extension*, *no orientation*.
 
 * **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.
+#### Framework of the paraxial approximation in Ray optics.
+
+* We use the concept of **light ray**, coming from *Ray optics*
+
+* **Ray optics** = **geometrical optics**
 
-* We use the concept of **light rays**, coming from *light rays optics*
+* *When optical systems are considered stigmatic*, their study is carried out in the framework of **paraxial ray optics**.
 
-* **light rays optics** = **geometrical optics**
+* **paraxial ray optics** = *paraxial approximation of Ray Optics* = *paraxial approximation of 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>
+* A *light pencil that diverges from a point source*, is modified by an imager and after emergence, *converges to a real image point* or *diverges from a virtual 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.
+    \- *All rays emerging from a point source* are deviated by the imager and *cross back on a corresponding real image point*, or *their extensions intersect at a corresponding virtual image point*.<br><br>
+$\Longrightarrow$  **knowledge of only two different rays** from a same 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)_
+Fig. 6a. Ray optics : 3 specific ray are specified (2 are sufficient) to locate the image point of any object point. In this figure the imager is a thin converging lens, and for this object position the corresponding image is real.
 
+![](imagerie-rays-optics-image-virtual.gif)<br>
+Fig. 6b. In this figure the imager is a thin diverging lens, and for this object position the corresponding image is virtual.
 
-### How is modeled and characterized a thin simple optical element in light rays optics?
+### How is modeled and characterized a thin simple optical element in paraxial rays optics?
 
 #### Thin simple optical element
 
@@ -168,11 +186,16 @@ $\Longrightarrow$ working in the **sectional view** corresponding to that plane
 * \- 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)
+
+![](objeto-imagen-bb.jpg)<br>
+Fig. 7a. The pencil coming from point source B intercepts the thin optical element. The corresponding emerging pencil converges on the point image B', which is a real image point because the light energy is concentrated at the point B'.
+![](object-image-conjugate-virtual.jpg)
+Fig. 7b. The pencil coming from point source B intercepts the thin optical element. The corresponding emerging pencil still diverges. But its extension shows that it diverges from an image point B' which is a virtual image point, because the light energy continues to disperse in space : no energy concentration in B'.
 
 * 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)
+![](plano-objeto-plan-image-bb.jpg)<br>
+Fig. 8.
 
 #### Coordinates to locate object and image points
 
@@ -188,15 +211,19 @@ $\Longrightarrow$ **(PO) and (PI) are conjugate planes**.
  \- 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)
+![](objeto-imagen-posicion-tamano-bb.jpg)<br>
+Fig. 9a.
+
+![](object-image-plan-conjugate-virtual.jpg)<br>
+Fig. 9b.
 
 #### Characterization of a thin simple optical element 
 
-**4 points ** * located on the optical axis* that characterize optical behavoir : **S , C, F and F**
+**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_)
+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*.
@@ -210,8 +237,6 @@ which define **3 important planes, perpendicular to the optical axis**
 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)*.
 
@@ -225,12 +250,12 @@ and **2 important algebraic distances**
 * **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)_
+Fig. 10. 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)_
+Fig. 11. Example : picture taken with a telephoto lens (telephoto lens is not a thin imager, but same image focal plane defintion).
 
 #### Object focal plane (P)
 
@@ -239,7 +264,7 @@ _example : picture taken with a telephoto lens (telephoto lens is not a thin ima
 * **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_)
+Fig. 12. Examples : the object can be the bulb of a lighthouse, or the film in a non-digital cinema projector (with quasi-parallel emerging beams in both cases, to illuminate front and far).
 
 ### How to determine the image given by a thin simple optical element ?
 
@@ -252,13 +277,15 @@ $\Longrightarrow$ dimensions perpendicular to optical axis $\ll$ dimensions alon
 
 * So **Accurate graphical study** $\Longrightarrow$ *greatly magnify scale perpendicular to optical axis*.
 
-![](objeto-imagen-posicion-tamano-bb.jpg)
+![](objeto-imagen-posicion-tamano-bb.jpg)<br>
+Fig. 13.
 
 ##### 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)
+![](objeto-imagen-3-rayos-bb.jpg)<br>
+Fig. 14.
 
 * **[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'.
@@ -306,7 +333,7 @@ From given point (object or image) **3 specific light rays can be drawn** ( only
 \- *thin spherical mirror equation*<br>
 \- *thin spherical refracting surface equation*<br>
 \- *thin lens equation*<br>
-(_will be demonstrated in level foothills_)
+(will be demonstrated in level foothills)
 
 ##### Distance of the conjugate point from optical axis
 
@@ -331,7 +358,7 @@ $\Longrightarrow$ **does not characterized the optical element** itself.
 
 ### How to characterize the action of an imager ?
 
-##### Characterization of an extended object
+#### 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>
@@ -340,7 +367,7 @@ $\Longrightarrow$ **does not characterized the optical element** itself.
 * 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
+#### 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>
@@ -349,22 +376,33 @@ $\Longrightarrow$ **does not characterized the optical element** itself.
 * 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 
+#### Characterization of the imager action on an extended object
 
 The imager gives an image of an object. The **characterization of imager action** depends on *how the object and the image are characterized*.
 
+##### Transverse magnification of the extended object
+
 * *Object and image* both **characterized by their algebraic transverse sizes** :<br>
 **$\Longrightarrow$** *imager action* characterized by the **transverse magnification $M_T$**<br><br>
 **$M_T=\dfrac{image\:size}{object\:size}=\dfrac{\overline{A'B'}}{\overline{AB}}$**
 
+##### Apparent magnification of the extended object
+
 * *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")_
 
+**$M_A=\dfrac{image\:apparent\:angle}{object\:apparent\:angle}=\dfrac{\overline{ \alpha '}}{\overline{ \alpha }}$**<br><br>
+or $M_A=\pm\dfrac{ \alpha '}{ \alpha }$, with sign + when erect image, sign - when inverted image.<br><br>
+
+![](apparent-magnitude-magnifying-glass-1018.gif)<br>
+Fig. 15. In this experiment, the observed object is located at a certain distance from the magnifying glass, as well as the eye of the observer. For these conditions of use, the magnifying glass gives a magnification of +2.5. There are optimum conditions for using the magnifying glass (see chapter "optical instruments").
+
+!!!! *BE CAREFUL*<br>
+!!!!An apparent angle depends on distance from the object or the image to the nodal point of the observing system (human eye or telephoto lens of a camera for example) $\Longrightarrow$ more accurate definitions of apparent angles will be necessary (see chapter "optical instruments").
+
+##### Longitudinal magnification of the extended object
 
 * *Object and image* both **characterized by their algebraic longitudinal sizes** :<br>
 **$\Longrightarrow$** *imager action* characterized by the **longitudinal magnification $M_L$**<br><br>
@@ -402,3 +440,10 @@ _an apparent angle depends on distance from nodal point $\Longrightarrow$ more a
 
 
 
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