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

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 u...

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Bibliographic Details
Published in:Discrete & computational geometry 2017-03, Vol.57 (2), p.431-469
Main Authors: Bobenko, Alexander I., Dimitrov, Nikolay, Sechelmann, Stefan
Format: Article
Language:English
Subjects:
Combinatorics
Computational geometry
Computational Mathematics and Numerical Analysis
Convexity
Curvature
Mathematical analysis
Mathematics
Mathematics and Statistics
Numerical analysis
Optimization
Optimization algorithms
Polyhedra
Theorems
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Summary: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.
ISSN:0179-5376
1432-0444
DOI:10.1007/s00454-016-9830-2