diff --git a/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/01.plane-refracting-surface/02.plane-refracting-surface-overview/cheatsheet.fr.md b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/01.plane-refracting-surface/02.plane-refracting-surface-overview/cheatsheet.fr.md
index b457c0f9b..19e9ce133 100644
--- a/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/01.plane-refracting-surface/02.plane-refracting-surface-overview/cheatsheet.fr.md
+++ b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/01.plane-refracting-surface/02.plane-refracting-surface-overview/cheatsheet.fr.md
@@ -3,6 +3,7 @@ title: 'Le dioptre sphérique, en approximation paraxiale : synthèse'
media_order: 'dioptre1ok.png,dioptre2ok.png,dioptre3ok.png,dioptre4ok.png'
---
+
+
+
+### Qu'est-ce qu'une interface réfractante ?
+
+#### Interface réfractante : description physique
+
+* **Interface séparant deux milieux transparents d'indices de réfraction différents.**.
+
+* **peut être trouvée dans la nature** :
+Exemples :
+\- une **interface réfractante plane** : la *surface plate et tranquille d'un lac*.
+\- une **interface réfractante sphérique** : un *aquarium boule*.
+
+
+Fig. 1. L'interface réfranctante sphérique d'un aquarium boule.
+
+* **apparaît dans la conception et modéisation de composants optiques ** :
+Exemples :
+\- une **vitre en verre** se décompose en *deux interface réfractantes planes* (air/verre, puis verre/air), séparées par l'épaisseur de la vitre.
+\- une **lentille** est la succession de *deux interfaces réfractantes courbes (souvent sphériques) consécutives* (air/verre, puis verre/air) qui
+présentent toutes deux une symétrie de révolution autour d'un même axe.
+
+#### Refracting interface versus refracting surface
+
+!!!! *DIFFICULT POINT* : One plane or spherical refracting interface has two different optical behaviors for image formation,
+is characterized by two different sets of parameters, depending of the direction of the light propagation.
+!!!!
+!!!!Consider a plane interface (a thick window whose thickness and effect can be neglected) separating air and water,
+and two twins (Thompson and Thomson) at equal distances on both sides of the interfaces (Fig. 2a).
+!!!!
+!!!! 
+!!!! Fig. 2a : The situation is not symmetrical.
+!!!!
+!!!! * When Thompson (in air) looks at Thomson (in water), the light propagates from Thomson to Thompson’s eyes.
+The fact is that Thompson sees the image of his brother closer than the real position of his brother (Fig. 2b)
+!!!!
+!!!! 
+!!!! Fig. 2b. Thompson sees his brother closer than his real position in water.
+!!!!
+!!!! * In the opposite situation, when Thomson (in water) looks at Thompson (in air),
+the light propagates from Thompson to Thomson’s eyes.
+And the fact is that Thomson sees the image of his brother farther away than his real position (Fig. 2c)
+!!!! (Strictly speaking, the eye of a fish should be considered in this situation, eyes well adapted to the vision in water,
+and in direct contact with water. If not, we should consider that the Thompson’s dive mask is filled with water,
+to have Thomson’s eyes in contact with water and not add another water/air refracting interface
+(that of the dive mask) on the path of the light).
+!!!!
+!!!! 
+!!!! Fig. 2c. Thomson sees in brother farther than his real position in air.
+!!!!
+!!!! All this can be predicted and calculated, but this example shows that this air/water plane
+refracting interface corresponds to two different plane refracting surfaces :
+!!!!
+!!!! * First case , refracting surface such as :
+!!!! \- refracting index of the medium of the incident light : $n_{inc} = n_{water} = 1.33$
+!!!! \- refracting index of the medium of the emergent light : $n_{eme} = n_{air} = 1$
+!!!!
+!!!! *¨ Second case , refracting surface such as :
+!!!! \- refracting index of the medium of the incident light : $n_{inc} = n_{air} = 1$
+!!!! \- refracting index of the medium of the emergent light : $n_{eme} = n_{water} = 1.33$.
+!!!!
+
+#### Difference in terminology between Spanish, French and English
+
+!!!! *BE CAREFUL* :
+!!!! In the same way as we use in English the single word "mirror" to qualify a "reflecting surface", in French is use the single word "dioptre" to qualify a "refracting surface".
+!!!! The term "dioptre" in English is a unit of mesure of the vergence of an optical system. In French, the same unit of measure is named "dioptrie".
+!!!! So keep in mind the following scheme :
+!!!!
+!!!! refracting surface : *EN : refracting surface* , *ES : superficie refractiva* , *FR : dioptre*.
+!!!! _A crystal ball forms a spherical refracting surface : un "dioptre sphérique" in French._
+!!!!
+!!!! unit of measure : *EN : dioptre* , *ES : dioptría* , *FR : dioptrie*.
+!!!! _My corrective lens for both eyes are 4 dioptres : "4 dioptries" in French._
+
+
+#### Non stigmatism of spherical refracting surfaces
+
+Ray tracing study of a **spherical refracting surface** :
+[Click here for geogebra animation](https://www.geogebra.org/material/iframe/id/x4hxqekd)
+
+* **At each impact point** of the rays upon the spherical refracting surface, the **Snell-Descartes relation applies**.
+
+
+
+* A spherical refracting surface is **not stigmatic** : The *rays (or their extensions)* originating *from a same object point* and that emerge from the surfac egenerally *do not converge towards an image point*.
+
+
+
+* **If we limit the aperture** of the spherical refracting surface so that only the rays
+meeting the surface near the vertex are refracted through the surface.
+
+
+
+* **and if** the object points remain close enough to the optical axis, so that the **angles of
+incidence and refraction remain small**, then for each object point an image point can be almost
+defined, and therefore the spherical refracting surface becomes *quasi-stigmatic*.
+
+
+
+
+#### Gauss conditions / paraxial approximation and quasi-stigmatism
+
+When spherical refracting surfaces are used under the following conditions, named **Gauss conditions** :
+\- All *incident rays lie close to the optical axis*
+\- The *angles of incidence and refraction are small*
+Then *the spherical refracting surfaces* can be considered *quasi-stigmatic*, and therefore they can be used to build optical images.
+
+Mathematically, when an angle $`\alpha`$ is small $`\alpha < or \approx 10 ^\circ`$, the following approximations can be made :
+$`sin(\alpha) \approx tan (\alpha) \approx \alpha`$, and $`cos(\alpha) \approx 1`$.
+
+*Geometrical optics limited to Gaussian conditions* is called *Gaussian optics* or *paraxial optics*.
+
+
+#### Thin spherical refracting surface
+
+We call **thin spherical refracting surface** a spherical refracting surface *used in the Gauss conditions*.
+
+
+### How is modeled a spherical refracting surface in paraxial optics ?
+
+
+#### Characterization of a spherical refracting surface
+
+* 2 distincts points : **vextex S** and **center of curvature C** on the optical axis,
+which defines $`\overline{SC}`$ : algebraic distance between vertex S and center C of curvature on optical axis.
+
+* 2 refractive index values :
+\- **$`n_{inc}`$ : refractive index of the medium of the incident light**.
+\- **$`n_{eme}`$ : refractive index of the medium of the emergent light**.
+
+* 1 arrow : indicates the *direction of light propagation*
+
+
+
+
+#### Analytical study
+
+
+* **Thin spherical refracting surface equation** = **conjuction equation** for a spherical refracting surface
+**$`\dfrac{n_{eme}}{\overline{SA_{ima}}}-\dfrac{n_{inc}}{\overline{SA_{obj}}}=\dfrac{n_{eme}-n_{inc}}{\overline{SC}}`$** (equ.1)
+
+* **Transverse magnification expression**
+ **$`\overline{M_T}=\dfrac{n_{inc}\cdot\overline{SA_{ima}}}{n_{eme}\cdot\overline{SA_{obj}}}`$**
+ (equ.2)
+You know $`\overline{SA_{obj}}`$, $`n_{inc}`$ and $`n_{eme}`$, you have previously calculated $`\overline{SA_{ima}}`$, so you can calculate $`\overline{M_T}`$ and deduced $`\overline{A_{ima}B_{ima}}`$.
+
+
+! *USEFUL* : The conjuction equation and the transverse magnification equation for a plane refracting
+!surface are obtained by rewriting these equations for a spherical refracting surface in the limit when
+!
+! $`|\overline{SC}|\longrightarrow\infty`$.
+! Then we get *for a plane refracting surface :*
+!
+! * *conjuction equation :* $`\dfrac{n_{fin}}{\overline{SA_{ima}}}-\dfrac{n_{ini}}{\overline{SA_{obj}}}=0`$ (equ.3)
+!
+! * *transverse magnification equation :* $`\dfrac{n_{ini}\cdot\overline{SA_{ima}}}{n_{fin}\cdot\overline{SA_{obj}}}`$
+ (equ.2, unchanged)
+! but (equ.3) gives $`\dfrac{\overline{SA_{ima}}}{\overline{SA_{obj}}}=\dfrac{n_{inc}}{n_{eme}}`$.
+! Copy this result into (equ.2) leads to $`\overline{M_T}=+1`$.
+
+
+#### Graphical study
+
+##### 1 - Determining object and image focal points
+
+Positions of object focal point F and image focal point F’ are easily obtained from the conjunction equation (equ. 1).
+
+* Image focal length $`\overline{OF'}`$ : $`\left(|\overline{OA_{obj}}|\rightarrow\infty\Rightarrow A_{ima}=F'\right)`$
+ (equ.1)$`\Longrightarrow\dfrac{n_{eme}}{\overline{SF'}}=\dfrac{n_{eme}-n_{inc}}{\overline{SC}}`$
+$`\Longrightarrow\overline{SF'}=\dfrac{n_{eme}\cdot\overline{SC}}{n_{eme}-n_{inc}}`$ (equ.4)
+
+* Object focal length $`\overline{OF}`$ : $`\left(|\overline{OA_{ima}}|\rightarrow\infty\Rightarrow A_{obj}=F\right)`$
+ (equ.1) $`\Longrightarrow-\dfrac{n_{inc}}{\overline{SF}}=\dfrac{n_{eme}-n_{inc}}{\overline{SC}}`$
+$`\Longrightarrow\overline{SF}=-\dfrac{n_{inc}\cdot\overline{SC}}{n_{eme}-n_{inc}}`$ (equ.5)
+
+!!!! *ADVISE* :
+!!!! Memory does not replace understanding. Do not memorise (equ.4) and (equ.5), but understand
+!!!! the definitions of the object and image focal points, and know how to find these two equations
+!!! from the conjuction equation for a spherical refracting surface.
+!!!!
+
+! *NOTE 1* :
+! An optical element being convergent when the image focal point is real,
+! so when $`\overline{OF}>0`$ (with optically axis positively oriented in the direction of the light propagation),
+! you can deduce from (equ.4) that is spherical refracting surface is convergent if and only if its center
+! of curvature C is in the mmedium of highest refractive index.
+!
+
+##### 2 - Thin spherical refracting surface representation
+
+* **Optical axis = revolution axis** of the refracting surface, positively **oriented** in the direction of
+propagation of the light (from the object towards the refracting surface)
+
+* Thin spherical refracting surface representation :
+\- **line segment**, perpendicular to the optical axis, centered on the axis with symbolic
+**indication of the direction of curvature** of the surface at its extremities.
+\- **vertex S**, that locates the refracting surface on the optical axis.
+\- **nodal point C = center of curvature**.
+\- **object focal point F and image focal point F’**.
+
+! *NOTE 2*
+! The direction of the curvature does not presume the convergent or divergent character
+! of the diopter. It also depends on the refractive index values on each side of the spherical
+! refracting surface. look at what happens to the incident ray parallel to the optical axis
+in Figures 3 and 4, and 5 and 6 below, and review NOTE 1.
+!
+
+#### Examples of graphical situations, with analytical results to train
+
+!!!! *IMPORTANT* :
+!!!! Even for only one of the following figures, the real or virtual character of the
+!!!! image may depend on the position of the object. This paragraph is only for you
+!!!! to understand how to determine the 3 rays that determine the image. It is
+!!!! important not to memorize these figures, which would be limiting, misleading
+!!!! and without interest.
+!!!!
+!!!! All the useful numerical values are given for each figure, making it possible
+!!!! also to check that you master the analytical study of each presented case.
+!!!!
+
+
+[Click here for geogebra animation](https://www.geogebra.org/material/iframe/id/gvkqgrpe)
+
+* with **real objects**
+
+
+Fig. 4.
+
+
+Fig. 5.
+
+
+Fig. 6.
+
+
+Fig. 7.
+
+* with **virtual objects**
+
+
+Fig. 8.
+
+
+Fig. 9.
+
+
+Fig. 10.
+
+
+Fig. 11.
+