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Spin Transformations of Discrete Surfaces

Crane, Keenan and Pinkall, Ulrich and Schröder, Peter (2011) Spin Transformations of Discrete Surfaces. ACM Transactions on Graphics, 30 (4). Art. No. 104. ISSN 0730-0301. doi:10.1145/1964921.1964999.

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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:Article
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URLURL TypeDescription
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
Caltech Center for Mathematics of InformationUNSPECIFIED
DFG Research Center MatheonUNSPECIFIED
DFG research unit Polyhedral SurfacesUNSPECIFIED
Subject Keywords:digital geometry processing, discrete differential geometry, geometric modeling, geometric editing, shape space deformation, conformal geometry, quaternions, spin geometry, Dirac operator
Issue or Number:4
Record Number:CaltechAUTHORS:20120104-114455813
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Official Citation: Spin transformations of discrete surfaces Keenan Crane, Ulrich Pinkall, Peter Schröder Article No.: 104 doi>10.1145/2010324.1964999
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:28651
Deposited By: Ruth Sustaita
Deposited On:04 Jan 2012 21:06
Last Modified:09 Nov 2021 16:59

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