Artistic Aspects of the Wigner Caustic and the Centre Symmetry Set

TL;DR

This paper explores the geometric and artistic properties of the Wigner caustic and Centre Symmetry Set of smooth planar curves, highlighting their singularities and applications across various scientific fields.

Contribution

It provides a detailed analysis of the artistic aspects and geometric structures of these singular sets, emphasizing their envelopes and cusp singularities.

Findings

Both sets generically admit cusp singularities.

They can be viewed as envelopes of special line families.

Applications span quantum physics, chaos theory, and convex geometry.

Abstract

The Wigner caustic and the Centre Symmetry Set of a closed smooth planar curve are known singular sets which generically admit only cusp singularities. Applications of these objects in semi-classical quantum physics, in chaos theory, in singularity theory, in convex geometry, have been studied since the 1970s until today. These sets can be viewed as envelopes of special families of lines and thanks to that they have many geometric artistic values.