Near ideal decompositions of ideal polygons

TL;DR

This paper proves that all ideal polygons can be decomposed into orthogeodesics of length at most approximately 2 log(n), providing a near-optimal decomposition method for ideal polygons.

Contribution

It offers a concise proof that ideal polygons admit orthogeodesic decompositions with lengths close to the theoretical minimum, advancing understanding of geometric decompositions.

Findings

All ideal polygons admit orthogeodesic decompositions.

Orthogeodesics in these decompositions have lengths at most ~2 log(n).

The length bound is roughly optimal.

Abstract

This article gives a short proof that all ideal polygons admit a short orthogeodesic decomposition. Specifically, all $n$-gons admit an orthogeodesic decomposition with orthogeodesics all of length at most $\sim 2 \log(n)$, and this is roughly optimal.