Update cheatsheet.en.md

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

@@ -105,21 +105,55 @@ 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
+! Then we get for a plane mirror : $`\overline{SA_{ima}}=\overline{SA_{obj}}`$ and
 ! $`\overline{M_T}=+1`$.
 
 ! *USEFUL 2° :<br>
 ! *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 :<br><br>
-! $`n_{eme}=-n_{inc}`$<br><br>
+! those of the spherical mirror**, with the following assumptions :<br>
+! $`n_{eme}=-n_{inc}`$<br>
 ! (to memorize : medium of incidence=medium of emergence, therefor same speed of light, but direction
-! of propagation reverses after reflection on the mirror)<br><br>
-!
+! of propagation reverses after reflection on the mirror)<br>
 ! are obtained by rewriting these two equations for a spherical refracting surface in the limit 
 ! when $`|\overline{SC}|\longrightarrow\infty`$.
 ! Then we get for a plane mirror :<br>
 ! $`\overline{SA_{ima}}=\overline{SA_{obj}}`$ and $`\overline{M_T}=+1`$
 
+##### 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 $`\overline{OF'}`$ : $`\left(|\overline{OA_{obj}}|\rightarrow\infty\Rightarrow A_{ima}=F'\right)`$<br><br>
+(equ.1) $`\Longrightarrow\dfrac{1}{\overline{SF'}}=\dfrac{2}{\overline{SC}}\Longrightarrow\overline{SF'}=\dfrac{\overline{SC}}{2}`$
+
+* Object focal length $`\overline{OF}`$ : $`\left(|\overline{OA_{ima}}|\rightarrow\infty\Rightarrow A_{obj}=F\right)`$<br><br>
+(equ.2) $`\Longrightarrow\dfrac{1}{\overline{SF}}=\dfrac{2}{\overline{SC}}\Longrightarrow\overline{SF}=\dfrac{\overline{SC}}{2}`$
+
+*2 - Thin spherical mirror representation*
+
+* **Optical axis = revolution axis** of the mirror, positively **oriented** in the direction of propagation of the incident light.
+
+* Thin spherical mirror equation :<br><br>
+\-**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.<br><br>
+\-**vertex S**, that locates the refracting surface on the optical axis;<br><br>
+\-**nodal point C = center of curvature**.<br><br>
+\-**object focal point F** and **image focal point F’**.
+
+##### Examples of graphical situations, with analytical results to train
+
+* with **real objects** 
+
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+