Isomorphism classes of cut loci for a cube

TL;DR

This paper classifies the different isomorphism classes of cut loci on a cube, providing a detailed partition of the face into regions with identical cut locus structures, supported by polynomial equations.

Contribution

It introduces a comprehensive classification of 177 distinct cut locus classes on a cube face, including polynomial equations for their boundaries.

Findings

177 distinct cut locus classes identified

60 connected open sets with constant cut locus class

Polynomial equations for boundary curves provided

Abstract

We prove that a face of a cube can be optimally partitioned into connected 193 sets on which the cut locus, or ridge tree, is constant up to isomorphism as a labeled graph. These are 60 connected open sets, curves bounding them, and intersection points of curves. Polynomial equations for the curves are provided. Sixteen pairs of sets support the same cut locus class. We present the 177 distinct cut locus classes.