Discrete & Computational Geometry

, Volume 48, Issue 3, pp 580–595 | Cite as

Affine Properties of Convex Equal-Area Polygons

  • Marcos Craizer
  • Ralph C. Teixeira
  • Moacyr A. H. B. da Silva
Article

Abstract

In this paper we discuss some affine properties of convex equal-area polygons, which are convex polygons such that all triangles formed by three consecutive vertices have the same area. Besides being able to approximate closed convex smooth curves almost uniformly with respect to affine length, convex equal-area polygons admit natural definitions of the usual affine differential geometry concepts, like affine normal and affine curvature. These definitions lead to discrete analogous to the six-vertex theorem and an affine isoperimetric inequality. One can also define discrete counterparts of the affine evolute, parallels and the affine distance symmetry set preserving many of the properties valid for smooth curves.

Keywords

Equal-area polygons Discrete six-vertex theorem Discrete isoperimetric inequality Discrete affine evolute Discrete affine distance symmetry set 

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Marcos Craizer
    • 1
  • Ralph C. Teixeira
    • 2
  • Moacyr A. H. B. da Silva
    • 3
  1. 1.Departamento de MatemáticaPUC-RioRio de JaneiroBrazil
  2. 2.Departamento de Matemática AplicadaUFFNiteróiBrazil
  3. 3.Centro de Matemática AplicadaFGVRio de JaneiroBrazil

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