Duality structures and discrete conformal variations of piecewise constant curvature surfaces

TL;DR

This paper introduces a comprehensive framework for analyzing discrete conformal structures on piecewise constant curvature surfaces, providing formulas for angle and curvature variations and classifying these structures across Euclidean, hyperbolic, and spherical geometries.

Contribution

It develops new formulas for angle and curvature derivatives in discrete conformal structures and offers a complete classification in various background geometries.

Findings

Formulas for derivatives of angles in each geometry

Identification of curvature functionals related to conformal variations

Complete classification of discrete conformal structures

Abstract

A piecewise constant curvature manifold is a triangulated manifold that is assigned a geometry by specifying lengths of edges and stipulating that for a chosen background geometry (Euclidean, hyperbolic, or spherical), each simplex has an isometric embedding into the background geometry with the chosen edge lengths. Additional structure is defined either by giving a geometric structure to the Poincare dual of the triangulation or by assigning a discrete metric, a way of assigning length to oriented edges. This notion leads to a notion of discrete conformal structure, generalizing the discrete conformal structures based on circle packings and their generalizations studied by Thurston and others. We define and analyze conformal variations of piecewise constant curvature 2-manifolds, giving particular attention to the variation of angles. We give formulas for the derivatives of angles in each background geometry, which yield formulas for the derivatives of curvatures. Our formulas allow us to identify particular curvature functionals associated with conformal variations. Finally, we provide a complete classification of discrete conformal structures in each of the background geometries.