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Discrete Conformal Mappings via Circle Patterns

Kharevych, Liliya and Springborn, Boris and Schröder, Peter (2006) Discrete Conformal Mappings via Circle Patterns. ACM Transactions on Graphics, 25 (2). pp. 412-438. ISSN 0730-0301. https://resolver.caltech.edu/CaltechAUTHORS:20110630-141124886

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Abstract

We introduce a novel method for the construction of discrete conformal mappings from surface meshes of arbitrary topology to the plane. Our approach is based on circle patterns, that is, arrangements of circles---one for each face---with prescribed intersection angles. Given these angles, the circle radii follow as the unique minimizer of a convex energy. The method supports very flexible boundary conditions ranging from free boundaries to control of the boundary shape via prescribed curvatures. Closed meshes of genus zero can be parameterized over the sphere. To parameterize higher genus meshes, we introduce cone singularities at designated vertices. The parameter domain is then a piecewise Euclidean surface. Cone singularities can also help to reduce the often very large area distortion of global conformal maps to moderate levels. Our method involves two optimization problems: a quadratic program and the unconstrained minimization of the circle pattern energy. The latter is a convex function of logarithmic radius variables with simple explicit expressions for gradient and Hessian. We demonstrate the versatility and performance of our algorithm with a variety of examples.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1145/1138450.1138461DOIUNSPECIFIED
http://portal.acm.org/citation.cfm?doid=1138450.1138461PublisherUNSPECIFIED
Additional Information:© 2006 ACM. Received July 2005; revised September 2005; accepted October 2005. This work was supported in part by NSF (DMS-0220905, DMS-0138458, ACI-0219979), DFG Research Center MATHEON “Mathematics for key technologies”, DOE (W-7405-ENG-48/B341492), Center for the Mathematics of Information, Alias, and Pixar. Special thanks to Alexander Bobenko, Mathieu Desbrun, Ilja Friedel, Nathan Litke, and Cici Koenig.
Funders:
Funding AgencyGrant Number
NSFDMS-0220905
NSFDMS-0138458
NSFACI-0219979
Department of Energy (DOE)W-7405-ENG-48/B341492
Center for the Mathematics of InformationUNSPECIFIED
AliasUNSPECIFIED
PixarUNSPECIFIED
DFG Research Center UNSPECIFIED
Subject Keywords:Conformal parameterizations, discrete analytic functions, discrete differential geometry, circle patterns, meshing, texture mapping
Issue or Number:2
Record Number:CaltechAUTHORS:20110630-141124886
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20110630-141124886
Official Citation:Discrete conformal mappings via circle patterns Liliya Kharevych, Boris Springborn, Peter Schröder Pages: 412 - 438 doi>10.1145/1138450.1138461
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:24285
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:30 Jun 2011 21:30
Last Modified:03 Oct 2019 02:54

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