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Spin transformations of discrete surfaces

Crane, Keenan and Pinkall, Ulrich and Schröder, Peter (2011) Spin transformations of discrete surfaces. In: ACM SIGGRAPH 2011 papers. ACM , New York, NY, Art. No. 104. ISBN 978-1-4503-0943-1.

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We introduce a new method for computing conformal transformations of triangle meshes in ℝ^3. Conformal maps are desirable in digital geometry processing because they do not exhibit shear, and therefore preserve texture fidelity as well as the quality of the mesh itself. Traditional discretizations consider maps into the complex plane, which are useful only for problems such as surface parameterization and planar shape deformation where the target surface is flat. We instead consider maps into the quaternions H, which allows us to work directly with surfaces sitting in ℝ^3. In particular, we introduce a quaternionic Dirac operator and use it to develop a novel integrability condition on conformal deformations. Our discretization of this condition results in a sparse linear system that is simple to build and can be used to efficiently edit surfaces by manipulating curvature and boundary data, as demonstrated via several mesh processing applications.

Item Type:Book Section
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Schröder, Peter0000-0002-0323-7674
Additional Information:© 2011 ACM. The authors thank Mirela Ben-Chen and Fabian Aiteanu for comparison data, Fernando de Goes for his Green Coordinates implementation, and Jessica Pfeilsticker for illuminating discussions on spin dynamics. Example meshes are courtesy of Autodesk, Luxology, 3D Universe, David Bommes, and Chris Legasse; cat clip art was created by Jon Phillips and Gerald Ganson. This research was partially funded by a Google PhD Fellowship, the Center for the Mathematics of Information at Caltech, the IAS at TU München, DFG Research Center Matheon, DFG Research Unit Polyhedral Surfaces and BMBF project GEOMEC.
Funding AgencyGrant Number
Google PhD FellowshipUNSPECIFIED
Center for the Mathematics of Information, CaltechUNSPECIFIED
Technical University of MunichUNSPECIFIED
Deutsche Forschungsgemeinschaft (DFG)UNSPECIFIED
Bundesministerium für Bildung und Forschung (BMBF)UNSPECIFIED
Subject Keywords:digital geometry processing, discrete differential geometry, geometric modeling, geometric editing, shape space deformation, conformal geometry, quaternions, spin geometry, Dirac operator
Record Number:CaltechAUTHORS:20161005-155130201
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Official Citation:Keenan Crane, Ulrich Pinkall, and Peter Schröder. 2011. Spin transformations of discrete surfaces. In ACM SIGGRAPH 2011 papers (SIGGRAPH '11), Hugues Hoppe (Ed.). ACM, New York, NY, USA, , Article 104 , 10 pages. DOI=
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:70899
Deposited On:05 Oct 2016 23:11
Last Modified:11 Nov 2021 04:36

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