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.
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| 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. | ||||||||||||||||||||
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| 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: | INVALID USER | ||||||||||||||||||||
| Deposited On: | 17 Nov 2016 00:59 | ||||||||||||||||||||
| Last Modified: | 09 Mar 2020 13:19 |
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