Duality of the Evolute-Involute Pair and Its Application

We studied the type of cusps on the caustic in the optical model that the coplanar point light source irradiates a singular planar curvilinear mirror in our last paper. In this paper, we give a oneto-one correspondence between the wavefront (orthotomic) and the caustic. According to this one-toone correspondence, we can derive the property of one of them from that of the other one. To prove it, we formulate the involutes of (n, m)-cusp curves and study their singular properties. Furthermore, we give a duality theorem for evolute-involute pairs. This theorem plays a crucial role in the one-to-one correspondence mentioned earlier.