Geodesic complexity of the octahedron, and an algorithm for cut loci on convex polyhedra

TL;DR

This paper establishes the geodesic complexity of the octahedron as four, introduces an algorithm for computing cut loci on convex polyhedra, and connects these to the space's geodesic complexity.

Contribution

It provides a new lower bound technique for geodesic complexity and an algorithm for cut loci on convex polyhedra, extending previous results to the octahedron.

Findings

Geodesic complexity of the octahedron is four.

Developed a correct algorithm for cut loci on convex polyhedra.

Extended prior work to include the octahedron, n-torus, and tetrahedron.

Abstract

The geodesic complexity of a length space $X$ quantifies the required number of case distinctions to continuously choose a shortest path connecting any given start and end point. We prove a local lower bound for the geodesic complexity of $X$ obtained by embedding simplices into $X\times X$. We additionally create and prove correctness of an algorithm to find cut loci on surfaces of convex polyhedra, as the structure of a space's cut loci is related to its geodesic complexity. We use these techniques to prove the geodesic complexity of the octahedron is four. Our method is inspired by earlier work of Recio-Mitter and Davis, and thus recovers their results on the geodesic complexity of the $n$-torus and the tetrahedron, respectively.