On the Development of the Intersection of a Plane with a Polytope

TL;DR

This paper investigates the geometric properties of the intersection curves formed when a plane cuts through a polytope, proving that such 'slice' curves are non-self-intersecting and introducing a generalized lemma to support this.

Contribution

It introduces a proof that slice curves on polytopes are non-self-intersecting and generalizes Cauchy's arm lemma for nonconvex planar chains.

Findings

Slice curves develop without self-intersection

Generalization of Cauchy's arm lemma for nonconvex chains

Theoretical foundation for geometric analysis of polytopes

Abstract

Define a ``slice'' curve as the intersection of a plane with the surface of a polytope, i.e., a convex polyhedron in three dimensions. We prove that a slice curve develops on a plane without self-intersection. The key tool used is a generalization of Cauchy's arm lemma to permit nonconvex ``openings'' of a planar convex chain.