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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. https://resolver.caltech.edu/CaltechAUTHORS:20161116-140825377

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Abstract

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
Related URLs:
URLURL TypeDescription
http://dx.doi.org/0.1145/1198555.1198665DOIChapter
http://dl.acm.org/citation.cfm?doid=1198555.1198665PublisherChapter
ORCID:
AuthorORCID
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.
Funders:
Funding AgencyGrant Number
NSFDMS-0220905
NSFDMS-0138458
NSFACI-0219979
Deutsche Forschungsgemeinschaft (DFG)UNSPECIFIED
Department of Energy (DOE)W-7405-ENG-48/B341492
nVidiaUNSPECIFIED
Center for Integrated Multiscale Modeling and SimulationUNSPECIFIED
AliasUNSPECIFIED
PixarUNSPECIFIED
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
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20161116-140825377
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=http://dx.doi.org/10.1145/1198555.1198665
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:72062
Collection:CaltechAUTHORS
Deposited By: Kristin Buxton
Deposited On:17 Nov 2016 00:59
Last Modified:09 Mar 2020 13:19

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