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.en.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.en.md
index 937570a64..80a3ea07f 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.en.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.en.md
@@ -3,22 +3,119 @@ title: 'Spherical refracting surface  : overview'
 media_order: dioptre-1.gif
 ---
 
-### What is a refracting surface ? 
+### What is a refracting interface ? 
 
-#### Spherical refracting surface : physical description
+#### Refracting surface : physical description
 
-* Local **spherical interface separating two transparent media of different refractive indices**.
+* **Interface separating two transparent media of different refractive indices**.
 
-* can be **foud in nature** : examples : when we look through the flat and quiet surface of a lake*, or when we look at something inside a fish ball aquarium.
+* can be **found in nature** :<br>
+Examples :<br>
+\- a **plane refracting surface** : the *flat and quiet surface of a lake*.
+\- a **spherical refracting surface** : a *fish ball aquarium*.
 
-<!-- fig 1 to add-→
-<br>
+![](spherical-refracting-surface-example-1.jpg)<br>
 Fig. 1. The spherical refracting interface of a fish ball aquarium.
 
-* **appears in the design and modeling of other optical elements** : examples:<br>
-\- a glass window pane is the combinaison of two parallel plane refracting interfaces (air/glass, then glass/air) that are rotationaly symmetrical around a same axis.
+* **appears in the design and modeling of other optical elements** :<br>
+Examples :<br>
+\- a **glass window pane** is the combinaison of *two parallel plane refracting interfaces* (air/glass, then glass/air), separated by the thickness of the pane.<br>
+\- a **lens** is composed of *two consecutive curved (often spherical) refracting interfaces* (air/glass, then glass/air) that are rotational symmetrical around a same axis.
+
+#### 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).
+!!!!
+!!!! ![](plane-refracting-surface-1.jpg)<br>
+!!!! 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)
+!!!!
+!!!! ![](plane-refracting-surface-2.jpg)<br>
+!!!! 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)<br>
+!!!! (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).
+!!!!
+!!!! ![](plane-refracting-surface-3.jpg)<br>
+!!!! 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 :<br>
+!!!!
+!!!! * First case , refracting surface such as :<br>
+!!!!   \- refracting index of the medium of the incident light : $n_{inc} = n_{water} = 1.33$<br>
+!!!!   \- refracting index of the medium of the emergent light : $n_{eme} = n_{air} = 1$<br>
+!!!!
+!!!! *¨ Second case , refracting surface such as :<br>
+!!!! \- refracting index of the medium of the incident light : $n_{inc} = n_{air} = 1$<br>
+!!!! \- refracting index of the medium of the emergent light : $n_{eme} = n_{water} = 1.33$.
+!!!!
+
+#### Non stigmatism of spherical refracting surfaces
+
+![](dioptre-spherique-snell-law)<br>
+Fig. 3. : In each point of the spherical refracting surface, the Snell-Descartes relation applies.
+
+![](dioptre-spherique-non-stigmatique-1)<br>
+Fig. 4. : A spherical refracting surface is not stigmatic: The rays (or their extensions) coming 
+from an object point generally do not converge towards an image point.
+
+![](dioptre-spherique-non-stigmatique-2)<br>
+Fig. 5a. : If we limit the opening of the spherical refracting surface so that only the rays 
+meeting the surface near the vertex are refracted through the surface.
+
+![](dioptre-spherique-non-stigmatique-3)<br>
+Fig. 5b. : and if the object points remain close 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** :<br>
+\- All *incident rays lie close to the optical axis*<br>
+\- The *angles of incidence and refraction are small*<br>
+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 :<br>
+$`sin(\alpha) \approx  tan (\alpha) \approx \alpha`$, and $`cos(\alpha) \approx 1`$.
+
+*Geometrical optics limited to Gaussian conditions* is called *Gaussian optics* or *paraxial optics*.
+
+#### 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 :<br>
+\- **$`n-{inc} : refractive index of the medium of the incident light**.<br>
+\- **$`n-{eme} : refractive index of the medium of the emergent light**.
+
+* 1 arrow : indicates the *direction of light propagation*
+
+#### Thin spherical refracting surface
+
+We call **thin spherical refracting surface** a spherical refracting surface *used in the Gauss conditions*.
+
+#### Analytical study (in paraxial optics)
+
+
+
+
+
+
+
 
-#### Spherical refracting interface versus 
 spherical refracting surface
 
 !!!! DIFFICULT  POINT:<br>