Poncelet Curves

TL;DR

This paper explores Poncelet curves, providing formulas to compute vertex curves and envelopes, and extends classical Poncelet theorem results to sets of curves with special polygonal properties.

Contribution

It introduces universal formulas for pairs of Poncelet curves and generalizes Poncelet's theorem to sets of curves with specific polygonal configurations.

Findings

Derived formulas for vertex curves and envelopes of Poncelet curves.

Extended Poncelet theorem to sets of curves with special polygons.

Analyzed Poncelet polygons that are equiangular or congruent.

Abstract

We examine pairs of closed plane curves that have the same closing property as two conic sections in Poncelet's porism. We show how the vertex curve can be computed for a given envelope and vice versa. Our formulas are universal in the sense that they produce all possible sufficiently regular pairs of such Poncelet curves. We arrive at similar results for sets of curves, analogous to the pencil of conic sections in the full Poncelet theorem. We also study the case of Poncelet curves that carry Poncelet polygons which are equiangular or even congruent.