Reflection from a spherical mirror

A spherical mirror intersects the optical axis at $(x=0,y=0)$. The mirror has a radius $R$ centered at $C$. For $R>0$ the interface is convex and for $R<0$ the interface is concave. Light rays leave an object $o$ on the left of the interface and are reflected at the interface at point $P$. to approximate a flat mirror.

Question

<p>Rays that make a small angle with the optical axis meet at an image point $i$. For <input type="button" value="R > 0" onclick="data.xs.value=-5; data.ys.value=0.5; data.R1.value=5;  data.rayphi.value =0; document.getElementById('point').checked = true;  update();" /> a virtual image appears behind the mirror where the extensions of the ray reflections meet. To someone looking at the mirror, the object appears to be coming from the point of the virtual image. The position $(x_i,y_i)$ of the image point is given by,</p>

$$\frac{1}{x_o}+\frac{1}{x_i}=\frac{2}{R},$$

$$ y_i = \frac{-y_oR}{2(x_o-R/2)}.$$

<p>The derivation of these equations assumes that the object is <input type="button" value="close to the optical axis" onclick="data.xs.value=-5; data.ys.value=0.5; data.R1.value=-5;  data.rayphi.value =0; document.getElementById('point').checked = true;  update();" /> so that the angle that the rays leaving the object make with the optical axis is small. The 'beam' setting only displays rays close to the optical axis where the approximation for the image point is generally satisfied. If the object is <input type="button" value="far from the optical axis" onclick="data.xs.value=-5; data.ys.value=2.5; data.R1.value=-5;  data.rayphi.value =0; document.getElementById('point').checked = true;  update();" />  the rays don't focus at a single image point.</p>






	Physik für Geodäsie 511.018 / Physik M 513.805