Duality between wavefronts and caustics

Abstract

Abstract There is a classical optical system: a planar curvilinear mirror illuminated by light rays coming from a coplanar point light source. An interesting and meaningful problem is to determine the focal points on the mirror corresponding to the cusps on the caustic. Due to the restriction of research on singular curves, this problem is still not very clear. In our last paper, we obtain a one-to-one correspondence between cusps on the orthotomic and cusps on the caustic. However, we know that vertexes on the orthotomic also induce cusps on the caustic. In this paper, we give their explicit correspondence and study the relation among mirror, orthotomic and caustic. This completes the work of the correspondence among mirror, wavefront (orthotomic) and caustic when the mirror has cusps and inflection points.