Discrete Calabi Flow: A Unified Conformal Parameterization Method

Abstract

Conformal parameterization for surfaces into various parameter domains is a fundamental task in computer graphics. Prior research on discrete Ricci flow provided us with promising inspirations from methods derived via Riemannian geometry, which is rigorous in theory and effective in practice. In this paper, we propose a unified conformal parameterization approach for turning triangle meshes into planar and spherical domains using discrete Calabi flow on piecewise linear metric. We incorporate edgeflipping surgery to guarantee convergence as well as other significant improvements including approximate Newton's method, optimal step-lengths, priority embedding and boundary customizing, which achieve better performance and functionality with robustness and accuracy.