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Discrete conformal mappings via circle patterns

Kharevych, Liliya and Springborn, Boris and Schröder, Peter (2005) Discrete conformal mappings via circle patterns. In: SIGGRAPH '05 ACM SIGGRAPH 2005 Courses. ACM , New York, NY, Ch. 6.

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We introduce a novel method for the construction of discrete conformal mappings from (regions of) embedded meshes to the plane. Our approach is based on circle patterns, i.e., 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 has two principal advantages over earlier approaches based on discrete harmonic mappings: (1) it supports very flexible boundary conditions ranging from natural boundaries to control of the boundary shape via prescribed curvatures; (2) the solution is based on a convex energy as a function of logarithmic radius variables with simple explicit expressions for gradients and Hessians, greatly facilitating robust and efficient numerical treatment. We demonstrate the versatility and performance of our algorithm with a variety of examples.

Item Type:Book Section
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URLURL TypeDescription
Schröder, Peter0000-0002-0323-7674
Additional Information:© 2005 ACM. This work was supported in part by NSF (DMS-0220905, DMS-0138458, ACI-0219979), DFG (Research Center MATHEON “Mathematics for Key Technologies,” Berlin), DOE (W-7405-ENG-48/B341492), nVidia, the Center for Integrated Multiscale Modeling and Simulation, Alias, and Pixar. Special thanks to Alexander Bobenko, Mathieu Desbrun, Ilja Friedel, and Cici Koenig.
Funding AgencyGrant Number
Deutsche Forschungsgemeinschaft (DFG)UNSPECIFIED
Department of Energy (DOE)W-7405-ENG-48/B341492
Center for Integrated Multiscale Modeling and SimulationUNSPECIFIED
Subject Keywords:Conformal parameterizations; discrete differential geometry; circle patterns; discrete analytic functions; meshing; texture mapping
Classification Code:G.1.0 [Numerical Analysis]: General— Numerical Algorithms; I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Curve,surface, solid and object representations; Geometric algorithms, languages, and systems
Record Number:CaltechAUTHORS:20161116-140825377
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Official Citation:Liliya Kharevych, Boris Springborn, and Peter Schröder. 2005. Discrete conformal mappings via circle patterns. In ACM SIGGRAPH 2005 Courses (SIGGRAPH '05), John Fujii (Ed.). ACM, New York, NY, USA, , Article 6 . DOI=
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:72062
Deposited On:17 Nov 2016 00:59
Last Modified:09 Mar 2020 13:19

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