Efficient and Robust Discrete Conformal Equivalence with Boundary

TL;DR

This paper presents an efficient algorithm for computing conformally equivalent metrics on discrete surfaces with boundaries, allowing prescribed curvature conditions and addressing boundary-specific challenges.

Contribution

It introduces a novel method that handles boundary conditions explicitly in discrete conformal mappings, extending previous theories to more general surface geometries.

Findings

Algorithm efficiently computes conformal metrics with boundary conditions.

Addresses stability issues in Delaunay triangulations with boundary.

Extends discrete conformal equivalence theory to surfaces with boundary.

Abstract

We describe an efficient algorithm to compute a conformally equivalent metric for a discrete surface, possibly with boundary, exhibiting prescribed Gaussian curvature at all interior vertices and prescribed geodesic curvature along the boundary. Our construction is based on the theory developed in [Gu et al. 2018; Springborn 2020], and in particular relies on results on hyperbolic Delaunay triangulations. Generality is achieved by considering the surface's intrinsic triangulation as a degree of freedom, and particular attention is paid to the proper treatment of surface boundaries. While via a double cover approach the boundary case can be reduced to the closed case quite naturally, the implied symmetry of the setting causes additional challenges related to stable Delaunay-critical configurations that we address explicitly in this work.