Duality of the Evolute-Involute Pair and Its Application

• Published in: IEEE access Vol. 7; pp. 72072 - 72075
• Main Authors: Zhang, Cuilian, Pei, Donghe
• Format: Journal Article
• Language: English
• Published: Piscataway IEEE 2019; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Duality theorem; Geometry; Involutes; Light sources; Mirrors; Optical imaging; optical propagation; Optical reflection; Optical surface waves; Wave fronts
• ISSN: 2169-3536; 2169-3536
• DOI: 10.1109/ACCESS.2019.2918188

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 one-to-one correspondence between the wavefront (orthotomic) and the caustic. According to this one-to-one correspondence, we can derive the property of one of them from that of the other one. To prove it, we formulate the involutes of <inline-formula> <tex-math notation="LaTeX">(n,\;m) </tex-math></inline-formula>-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.