The distance from a point to its opposite along the surface of a box

TL;DR

This paper investigates the shortest surface path between a point and its opposite on a rectangular box, deriving algebraic formulas and classifying regions based on the path's face traversal.

Contribution

It introduces a method to determine minimal surface distances on a box by partitioning face regions and deriving explicit algebraic formulas based on box dimensions.

Findings

Shortest path is always a straight line in some planar flattening.

Partitioning of the face depends on which faces the path crosses.

Explicit algebraic formulas for distance are derived based on dimensions.

Abstract

Given a point (the "spider") on a rectangular box, we would like to find the minimal distance along the surface to its opposite point (the "fly" - the reflection of the spider across the center of the box). Without loss of generality, we can assume that the box has dimensions $1\times a\times b$ with the spider on one of the $1\times a$ faces (with $a\leq 1$). The shortest path between the points is always a line segment for some planar flattening of the box by cutting along edges. We then partition the $1\times a$ face into regions, depending on which faces this path traverses. This choice of faces determines an algebraic distance formula in terms of $a$, $b$, and suitable coordinates imposed on the face. We then partition the set of pairs $(a,b)$ by homeomorphism of the borders of the $1\times a$ face's regions and a labeling of these regions.