Wang, Ke and Wei, Wei and Tong, Yiying and Desbrun, Mathieu and Schröder, Peter
(2006)
*Edge Subdivision Schemes and the Construction of Smooth Vector Fields.*
ACM Transactions on Graphics, 25
(3).
pp. 1041-1048.
ISSN 0730-0301.
doi:10.1145/1179352.1141991.
https://resolver.caltech.edu/CaltechAUTHORS:20110804-135119206

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## Abstract

Vertex- and face-based subdivision schemes are now routinely used in geometric modeling and computational science, and their primal/dual relationships are well studied. In this paper, we interpret these schemes as defining bases for discrete differential 0- resp. 2-forms, and complete the picture by introducing edge-based subdivision schemes to construct the missing bases for discrete differential 1-forms. Such subdivision schemes map scalar coefficients on edges from the coarse to the refined mesh and are intrinsic to the surface. Our construction is based on treating vertex-, edge-, and face-based subdivision schemes as a joint triple and enforcing that subdivision commutes with the topological exterior derivative. We demonstrate our construction for the case of arbitrary topology triangle meshes. Using Loop's scheme for 0-forms and generalized half-box splines for 2-forms results in a unique generalized spline scheme for 1-forms, easily incorporated into standard subdivision surface codes. We also provide corresponding boundary stencils. Once a metric is supplied, the scalar 1-form coefficients define a smooth tangent vector field on the underlying subdivision surface. Design of tangent vector fields is made particularly easy with this machinery as we demonstrate.

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Additional Information: | © 2006 Association for Computing Machinery, Inc. This research has been supported in part by NSF (CCF-0528101, CCR-0133983, and ITR DMS-0453145), DOE (W-7405-ENG-48/B341492 and DE-FG02-04ER25657), the Caltech Center for Mathematics of Information, nVidia, and Autodesk. | ||||||||||||||||||

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Subject Keywords: | Subdivision; Discrete Exterior Calculus; Discrete Differential Geometry; vector fields; smooth surface modeling | ||||||||||||||||||

Issue or Number: | 3 | ||||||||||||||||||

DOI: | 10.1145/1179352.1141991 | ||||||||||||||||||

Record Number: | CaltechAUTHORS:20110804-135119206 | ||||||||||||||||||

Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20110804-135119206 | ||||||||||||||||||

Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||||||||||||||

ID Code: | 24691 | ||||||||||||||||||

Collection: | CaltechAUTHORS | ||||||||||||||||||

Deposited By: | INVALID USER | ||||||||||||||||||

Deposited On: | 04 Aug 2011 22:44 | ||||||||||||||||||

Last Modified: | 09 Nov 2021 16:25 |

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