Envelopes of lines, unfoldings and breaking symmetry

TL;DR

This paper extends the classical envelopes of chords in a circle by allowing rational slopes, using two concentric circles, and breaking symmetry to explore complex singularities like swallowtails and butterflies.

Contribution

It generalizes envelope formulas to rational slopes, introduces two-circle configurations, and employs symmetry breaking to study higher singularities in a unified framework.

Findings

Formulas for cusps, points at infinity, and self-intersections with rational m.

Identification of higher singularities such as swallowtails and butterflies.

Symmetry breaking enables versal unfolding of complex singularities.

Abstract

We generalise the well-known ``embroidery'' envelopes of chords joining points at angles $t$ and $mt$ of a single circle in several ways. Firstly we allow $m$ to be rational (possibly negative) instead of integral, finding formulas for the number of cusps, points at infinity and self-intersections of these envelopes. Secondly we use two concentric circles instead of one, taking the chords to join a point of one circle to a point of the other. This construction allows the formation of different (higher) singularities -- not just simple cusps -- which however do not reveal their full inner structure when changing the radius of one of the circles. For this we need to break some of the symmetry and move the center of one of the circles as well as its radius. This permits the higher singularities, including swallowtails and butterflies, to be ``versally unfolded'' in the language of singularity theory.