Generalized Antiorthotomics of (n, m)-Cusp Curves

In optics, light rays emitted from a light source form a wavefront (orthotomic) upon reflection by a mirror. The mirror is referred to as an antiorthotomic of the orthotomic. The investigation of the relationship between orthotomics and antiorthotomics constitutes an interesting problem in physics. However, the study becomes ambiguous when the orthotomic exhibits singular points. In this paper, we define generalized antiorthotomics of (n, m)-cusp curves in the Euclidean plane by using the singular curve theory. We demonstrate that the singular points of the generalized antiorthotomic sweep out the evolute of the (n, m)-cusp curve. We also investigate the behavior and singular characteristics of the antiorthotomic of the (n, m)-cusp curve. Moreover, we define parallels for (n, m)-cusp curves and reveal the relationship between parallels and generalized antiorthotomics. Finally, repeated antiorthotomics are studied, which is useful for identifying the characteristics of (n, m)-cusp curves.