Discrete Uniformization of Polyhedral Surfaces with Non-positive Curvature and Branched Covers over the Sphere via Hyper-ideal Circle Patterns

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• 作者：Alexander I. Bobenko ; Nikolay Dimitrov…
• 关键词：Hyper ; ideal circle pattern ; Discrete uniformization ; Discrete conformal map ; Branched cover ; Polyhedral surface ; Variational principle
• 刊名：Discrete & Computational Geometry
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：57
• 期：2
• 页码：431-469
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Combinatorics; Computational Mathematics and Numerical Analysis;
• 出版者：Springer US
• ISSN：1432-0444
• 卷排序：57

文摘

With the help of hyper-ideal circle pattern theory, we develop a discrete version of the classical uniformization theorems for closed polyhedral surfaces with non-positive curvature and for surfaces represented as finite branched covers over the Riemann sphere. We show that in these cases discrete uniformization via hyper-ideal circle patterns always exists and is unique. We also propose a numerical algorithm, utilizing convex optimization, that constructs the desired discrete uniformization.