The Four-Vertex Theorem, The Evolute, and The Decomposition of Polygons

2009/06/12 by Wiktor J. Mogilski, Mogilski, Wiktor J.
Computer Science · Engineering · Mathematics · #51F99 #Advanced Numerical Analysis Techniques #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #Digital Image Processing Techniques #FOS: Mathematics #Metric Geometry (math.MG) #math.CO #math.MG #msc:51F99

paper · pdf · doi:10.48550/arxiv.0906.2388

60 pages, 46 figures

openalex publication_date 2009/06/12 · arxiv created 2009/06/15 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Four-Vertex Theorem has been of interest ever since a discrete version appeared in 1813 due to Cauchy. Up until now, there have been many different versions of this theorem, both for discrete cases and smooth cases. In 2004, an approach relating the discrete Four-Vertex Theorem to the evolute was published, and here we will give an overview of this paper. We then will define the notion of the decomposition of polygons, and derive some new results about how this notion affects various types of extremality. We will see that from our fresh results we can easily derive discrete Four-Vertex Theorems.

Citations

• The Four Vertex Theorem and its Converse

Related

• Polygon Vertex Extremality and Decomposition of Polygons
• A Discrete Four Vertex Theorem for Hyperbolic Polygons
• New four-vertex type theorems for spherical polygons
• Projective geometry of polygons and discrete 4-vertex and 6-vertex theorems
• The Four Vertex Theorem and its Converse