Evolutes of (n, m)-cusp curves and application in optical system

The focal properties of generic curvilinear planar mirrors illuminated by a coplanar light source are very clear. However, the case is vague when the mirror has singularities. In this paper, we define the evolutes of (n, m)-cusp curves in Euclidean 2-space. The (n, m)-cusp curve is a generalization of regular curves which might have singularities. Furthermore, we study the behavior and the singular property of the evolute at the point corresponding to the (n, m)-cusp of the base curve. As an application, we study the caustic of the curvilinear planar mirror which is illuminated by a coplanar light source when the mirror is an (n, m)-cusp curve.