The Geometry of the Secant Caustic of a Planar Curve

TL;DR

This paper explores the geometric characteristics of the secant caustic of planar curves, focusing on branch structure, cusps, and inflection points, with detailed analysis of rosettes.

Contribution

It provides a detailed analysis of the geometric properties of the secant caustic, including branch count, cusp parity, and inflection points, especially for rosettes.

Findings

Number of branches of the secant caustic determined

Parity of cusps in secant caustic analyzed

Inflexion points in each branch characterized

Abstract

The secant caustic of a planar curve $M$ is the image of the singular set of the secant map of $M$. We analyse the geometrical properties of the secant caustic of a planar curve, i.e. the number of branches of the secant caustic, the parity of the number of cusps and the number of inflexion points in each branch of this set. In particular, we investigate in detail some of the geometrical properties of the secant caustic of a rosette, i.e. a smooth regular oriented closed curve with non-vanishing curvature.