Poncelet Plectra: Harmonious Curves in Cosine Space

TL;DR

This paper explores the geometric properties of Poncelet N-gons and related triangles in confocal elliptic billiards, revealing conserved cosine quantities and their geometric curves in both planar and spherical contexts.

Contribution

It uncovers new relationships between Poncelet polygons, their affine images, and cosine-based invariants, including the sweeping of common curves in different geometric spaces.

Findings

Poncelet N-gons conserve the sum of cosines of internal angles.

Triangles excentral to confocal families share conserved cosine products.

Cosine triples sweep the same planar and spherical curves.

Abstract

It has been shown that the family of Poncelet N-gons in the confocal pair (elliptic billiard) conserves the sum of cosines of its internal angles. Curiously, this quantity is equal to the sum of cosines conserved by its affine image where the caustic is a circle. We show that furthermore, (i) when N=3, the cosine triples of both families sweep the same planar curve: an equilateral cubic resembling a plectrum (guitar pick). We also show that (ii) the family of triangles excentral to the confocal family conserves the same product of cosines as the one conserved by its affine image inscribed in a circle; and that (iii) cosine triples of both families sweep the same spherical curve. When the triple of log-cosines is considered, this curve becomes a planar, plectrum-shaped curve, rounder than the one swept by its parent confocal family.