Chern, Albert and Pinkall, Ulrich and Schröder, Peter (2015) Close-to-conformal deformations of volumes. ACM Transactions on Graphics, 34 (4). Art. No. 56. ISSN 0730-0301. doi:10.1145/2766916. https://resolver.caltech.edu/CaltechAUTHORS:20150814-145256783
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Abstract
Conformal deformations are infinitesimal scale-rotations, which can be parameterized by quaternions. The condition that such a quaternion field gives rise to a conformal deformation is nonlinear and in any case only admits Möbius transformations as solutions. We propose a particular decoupling of scaling and rotation which allows us to find near to conformal deformations as minimizers of a quadratic, convex Dirichlet energy. Applied to tetrahedral meshes we find deformations with low quasiconformal distortion as the principal eigenvector of a (quaternionic) Laplace matrix. The resulting algorithms can be implemented with highly optimized standard linear algebra libraries and yield deformations comparable in quality to far more expensive approaches.
| Item Type: | Article | |||||||||
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| Additional Information: | © 2015 Copyright is held by the owner/author(s). Publication rights licensed to ACM. The work reported here was supported in part by ONR Award N00014-11-1002, the DFG Collaborative Research Center TRR 109, “Discretization in Geometry and Dynamics,” and a software donation from Side Effects Software. We are grateful to Houman Owhadi and Chiu-Yen Kao for generously sharing their knowledge; Kovalsky and Lipman for providing code from [Kovalsky et al. 2014]; and Paillé for the models used in [Paillé and Poulin 2012]. Last but not least the detailed reviewer feedback helped us greatly improve the paper. | |||||||||
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| Subject Keywords: | discrete differential geometry, digital geometry processing, conformal metric, deformations, geometric modeling | |||||||||
| Issue or Number: | 4 | |||||||||
| Classification Code: | CR Categories: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Geometric algorithms, languages, and systems | |||||||||
| DOI: | 10.1145/2766916 | |||||||||
| Record Number: | CaltechAUTHORS:20150814-145256783 | |||||||||
| Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20150814-145256783 | |||||||||
| Official Citation: | Chern, A., Pinkall, U., Schröder, P. 2015. Close-to-Conformal Deformation of Volumes. ACM Trans. Graph. 34, 4, Article 56 (August 2015), 13 pages. DOI = 10.1145/2766916 http://doi.acm.org/10.1145/2766916. | |||||||||
| Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | |||||||||
| ID Code: | 59563 | |||||||||
| Collection: | CaltechAUTHORS | |||||||||
| Deposited By: | George Porter | |||||||||
| Deposited On: | 14 Aug 2015 23:01 | |||||||||
| Last Modified: | 10 Nov 2021 22:22 |
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