A variational principle for cyclic polygons with prescribed edge lengths

TL;DR

This paper introduces a variational approach to prove the existence and uniqueness of cyclic polygons with given edge lengths across Euclidean, spherical, hyperbolic, and spacetime geometries, unifying several classical theorems.

Contribution

It presents a new variational proof for cyclic polygons with prescribed sides, extending classical results to spherical, hyperbolic, and spacetime geometries.

Findings

Proof of existence and uniqueness in Euclidean geometry.

Extension of the theorem to spherical and hyperbolic geometries.

Application to polygons in 1+1 spacetime.

Abstract

We provide a new proof of the elementary geometric theorem on the existence and uniqueness of cyclic polygons with prescribed side lengths. The proof is based on a variational principle involving the central angles of the polygon as variables. The uniqueness follows from the concavity of the target function. The existence proof relies on a fundamental inequality of information theory. We also provide proofs for the corresponding theorems of spherical and hyperbolic geometry (and, as a byproduct, in $1+1$ spacetime). The spherical theorem is reduced to the euclidean one. The proof of the hyperbolic theorem treats three cases separately: Only the case of polygons inscribed in compact circles can be reduced to the euclidean theorem. For the other two cases, polygons inscribed in horocycles and hypercycles, we provide separate arguments. The hypercycle case also proves the theorem for "cyclic" polygons in $1+1$ spacetime.