diff --git a/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/02.mirror/02.new-course-overview/cheatsheet.en.md b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/02.mirror/02.new-course-overview/cheatsheet.en.md
index f1d554312..494508cad 100644
--- a/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/02.mirror/02.new-course-overview/cheatsheet.en.md
+++ b/01.curriculum/01.physics-chemistry-biology/02.Niv2/04.optics/04.use-of-basic-optical-elements/02.mirror/02.new-course-overview/cheatsheet.en.md
@@ -27,16 +27,15 @@ intensity per the total of the incident light intensity), the surface is
##### Interest in optics
-* **One of the most importante simple optical component** that is used *alone or
-* combined in a series in most optical instruments* : some telephotos,
-* reflecting telescopes.
+* **One of the most importante simple optical component** that is used *alone or combined in a series in most optical instruments* :
+some telephotos, reflecting telescopes.
#### Why to study plane and spherical mirrors?
* **Plane and spherical mirrors** are the *most technically easy to realize*,
so they are the *most common and cheap*.
* In paraxial optics, the optical properties of a **plane mirror** are those
-of a *spherical mirror of infinite radius of curvature*.
+of a *spherical mirror whose radius of curvature tends towards infinity*.
Plane mirror, concave and convex spherical mirror
@@ -47,42 +46,40 @@ Fig. 1. a) plane b) concave c) convex mirrors
##### Perfect stigmatism of the plane mirror
* A plane mirror is **perfectly stigmatic**.
-* Object and image are symmetrical on both side of the surface of the plane mirror.
-* A real object gives a virtual image.
A virtual object gives a real image.
+* Object and image are symmetrical on both side of the surface of the plane mirror.
+$`\Longrightarrow`$ A real object gives a virtual image.
+ nbsp; A virtual object gives a real image.
##### Non stigmatism of the spherical mirror
* In each point of the spherical mirror, the law of reflection applies.
-* A spherical mirror is not stigmatic: The rays (or their extensions)
-* coming from an object point generally do not converge towards an image
-* point (see Fig. 2.)
-*
+* A spherical mirror is not stigmatic: The rays (or their extensions) coming from an object point generally do not converge towards an image point (see Fig. 2.)
+* A spherical mirrors with a limited aperture (see the angle $`\alpha`$ (rad) lower on Fig. 3. and 4.) and used so that
+angles of incidence remain small (see Fig. 4.) become quasi-stigmatic.
+
Fig. 2. Non stigmatism of a convexe mirror.
-Fig. 3. But when we limit the aperture of the mirror,
+Fig. 3. But when we limit the aperture of the mirror
-Fig. 4 . and limit the conditions of use to small angles of incidence and
-refraction are small, then a point image can be defined : the mirror becomes
+Fig. 4 . and limit the conditions of use to small angles of incidence, then a image point can almost be defined : the mirror becomes
quasi-stigmatic.
-* Spherical mirrors with a limited aperture (see Fig. 3.) and used so that
-angles of incense and emergence remain small (see Fig. 4.), become quasi-stigmatic.
##### Gauss conditions / paraxial approximation and quasi-stigmatism
-* When spherical refracting surfaces are used under the following conditions, named **Gauss conditions** :
-\- The *angles of incidence and refraction are small*
-(the rays are slightly inclined on the optical axis, and intercept the spherical surface in the
+* When spherical mirrors are used under the following conditions, named **Gauss conditions** :
+\- The *angles of incidence are small*
+(the rays are slightly inclined on the optical axis, and intercept the spherical mirror in the
vicinity of its vertex),
-then the spherical refracting surfaces can be considered *quasi- stigmatic*, and therefore they
+then the spherical mirrors 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
+* Mathematically, when an angle $`i`$ is small ($`i < or \approx 10 ^\circ`$), the following
approximations can be made :
-$`sin(\alpha) \approx tan (\alpha) \approx \alpha`$ (rad), et $`cos(\alpha) \approx 1`$.
+$`sin(i) \approx tan (i) \approx i`$ (rad), et $`cos(i) \approx 1`$.
* Geometrical optics limited to Gaussian conditions is called **Gaussian optical** or **paraxial optics**.
@@ -105,19 +102,17 @@ then $`\overline{M_T}`$ with (equ.2), and deduce $`\overline{A_{ima}B_{ima}}`$.
! The conjunction equation and the transverse magnification equation for a plane mirror
! are obtained by rewriting these two equations for a spherical mirror in the limit when
! $`|\overline{SC}|\longrightarrow\infty`$.
-! Then we get for a plane mirror : $`\overline{SA_{ima}}=\overline{SA_{obj}}`$ and
-! $`\overline{M_T}=+1`$.
+! Then we get for a plane mirror : $`\overline{SA_{ima}}= - \overline{SA_{obj}}`$ and $`\overline{M_T}=+1`$.
! *USEFUL 2* :
-! *You can find* the conjunction and the transverse magnification *equations for a plane mirror directly from
-! those of the spherical mirror*, with the following assumptions :
-! $`n_{eme}=-n_{inc}`$
+! *You can find* the conjunction and the transverse magnification *equations for a plane or spherical mirror as well as for a plane refracting surface directly from
+! those of the spherical refracting surface*, with the following assumptions :
+! - to go from refracting surface to mirror : $`n_{eme}=-n_{inc}`$
! (to memorize : medium of incidence=medium of emergence, therefor same speed of light, but direction
! of propagation reverses after reflection on the mirror)
-! are obtained by rewriting these two equations for a spherical refracting surface in the limit
-! when $`|\overline{SC}|\longrightarrow\infty`$.
+! - to go from spherical to plane : $`|\overline{SC}|\longrightarrow\infty`$.
! Then we get for a plane mirror :
-! $`\overline{SA_{ima}}=\overline{SA_{obj}}`$ and $`\overline{M_T}=+1`$
+! $`\overline{SA_{ima}}= - \overline{SA_{obj}}`$ and $`\overline{M_T}=+1`$
##### Graphical study
@@ -140,7 +135,7 @@ equation (equ. 1).
\-**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, and *dark or hatched area on the non-reflective
side* of the mirror.
-\-**vertex S**, that locates the refracting surface on the optical axis;
+\-**vertex S**, that indicates the position of the mirror along the optical axis;
\-**nodal point C = center of curvature**.
\-**object focal point F** and **image focal point F’**.