As mentioned previously the images of the target focal point can be used to find the reflection points on a mirror. Expressed alternatively this is where the line from the source to the a focal image intersects the plane of the mirror. We compute images and successive images using the reflection matrix operators. Upper indices in brackets can be used to designated the alternating sequence of reflections.
The intersection x̄ can be found by projecting the ray between point source x̄₀ and the focal image f̄ onto the mirror. If the the order of a double reflection of a ray is mirror 1 followed by mirror 2 one needs to use image f⁽²¹⁾ for the first intersection and f⁽²⁾ for the second reflection. The sequence of reflection for f⁽²¹⁾ is a reflection of the focal point f through mirror 2 followed by a second through mirror 1.
Using this method we can obtain a sequence of multiple reflections.
Plotting the data gives a better picture of what's happening.
An observer at the focal point on the right will see multiple images of the luminous source on the left with the directions indicated by the directions of the converging rays and a distance equal to the total path length the reflection segments.
Please note that the path length or "action" is a local minimum, i.e., when compared with infinitesimally neighboring rays.
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