The Geometry of the Centre Symmetry Set of a Planar Curve

TL;DR

This paper investigates the global geometric properties of the Centre Symmetry Set of planar curves, focusing on singularities and asymptotes, to deepen understanding of its structure and features.

Contribution

It provides new insights into the global geometry of the Centre Symmetry Set, including the enumeration of singularities and asymptotes for planar curves.

Findings

Identifies the number of singularities of the Centre Symmetry Set.

Determines the asymptotic behavior of the set.

Provides classifications of geometric configurations.

Abstract

The Centre Symmetry Set of a planar curve $M$ is the envelope of affine chords of $M$, i.e. the lines joining points on $M$ with parallel tangent lines. In this paper we study global geometrical properties of this set including the number of singularities and the number of asymptotes.