Discrete Conformal Deformation: Algorithm and Experiments

Jian Sun, Tianqi Wu, Xianfeng Gu, Feng Luo

Abstract: In this paper, we introduce the definition of discrete conformality for triangulated surfaces with flat cone metrics and describe an algorithm for solving the problem of prescribing curvature, which is to deform the metric discrete conformally so that the curvature of the resulting metric coincides with the prescribed curvature. We explicitly construct a discrete conformal map between the input triangulated surface and the deformed triangulated surface. Our algorithm can handle a surface with any topology, with or without boundary, and can find a deformed metric for any prescribed curvature satisfying the Gauss--Bonnet formula. In addition, we present the numerical examples to show the convergence of our discrete conformality and to demonstrate the efficiency and the robustness of our algorithm.