Duality structures and discrete conformal variations of piecewise constant curvature surfaces

David Glickenstein, Joseph Thomas

Research output: Contribution to journalArticle

Abstract

A piecewise constant curvature manifold is a triangulated manifold that is assigned a geometry by specifying lengths of edges and stipulating the simplex has an isometric embedding into a constant curvature background geometry (Euclidean, hyperbolic, or spherical) with the specified edge lengths. Additional geometric structure leads to a notion of discrete conformal structure, generalizing circle packings and their generalizations as studied by Thurston and others. We analyze discrete conformal variations of piecewise constant curvature 2-manifolds, giving particular attention to the variation of angles. Formulas are derived for the derivatives of angles in each background geometry, which yield formulas for the derivatives of curvatures and to curvature functionals. Finally, we provide a complete classification of possible definitions of discrete conformal structures in each of the background geometries.

Original languageEnglish (US)
Pages (from-to)250-278
Number of pages29
JournalAdvances in Mathematics
Volume320
DOIs
StatePublished - Nov 7 2017

Keywords

  • Circle packing
  • Conformal
  • Duality
  • Euclidean geometry
  • Hyperbolic geometry

ASJC Scopus subject areas

  • Mathematics(all)

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