Cauchy's Arm Lemma on a Growing Sphere

2008/04/07 by Zachary Abel, Abel, Zachary, D. G. Charlton +19 · 1 citation
Computer Science · Mathematics · #Algebraic and Geometric Analysis #Computational Geometry (cs.CG) #FOS: Computer and information sciences #History and Theory of Mathematics #Mathematical and Theoretical Analysis #cs.CG

paper · pdf · doi:10.48550/arxiv.0804.0986

arxiv created 2008/04/07 · openalex publication_date 2008/04/07 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We propose a variant of Cauchy's Lemma, proving that when a convex chain on one sphere is redrawn (with the same lengths and angles) on a larger sphere, the distance between its endpoints increases. The main focus of this work is a comparison of three alternate proofs, to show the links between Toponogov's Comparison Theorem, Legendre's Theorem and Cauchy's Arm Lemma.

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• Source Unfoldings of Convex Polyhedra via Certain Closed Curves

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