Article

Edge subdivision schemes and the construction of smooth vector fields

Authors:

Mathieu Desbrun,

Peter SchröderAuthors Info & Claims

SIGGRAPH '06: ACM SIGGRAPH 2006 Papers

Pages 1041 - 1048

https://doi.org/10.1145/1179352.1141991

Published: 01 July 2006 Publication History

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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Cited By

Gross BAtzberger P(2018)Spectral Numerical Exterior Calculus Methods for Differential Equations on Radial ManifoldsJournal of Scientific Computing10.1007/s10915-017-0617-276:1(145-165)Online publication date: 1-Jul-2018
https://dl.acm.org/doi/10.1007/s10915-017-0617-2
Zhang E(2012)Tensor Field Design: Algorithms and ApplicationsNew Developments in the Visualization and Processing of Tensor Fields10.1007/978-3-642-27343-8_6(111-133)Online publication date: 9-Jul-2012
https://doi.org/10.1007/978-3-642-27343-8_6
Ray NVallet BLi WLévy B(2008)N-symmetry direction field designACM Transactions on Graphics10.1145/1356682.135668327:2(1-13)Online publication date: 8-May-2008
https://dl.acm.org/doi/10.1145/1356682.1356683
Show More Cited By

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Edge subdivision schemes and the construction of smooth vector fields

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cover image ACM Conferences

SIGGRAPH '06: ACM SIGGRAPH 2006 Papers

July 2006

742 pages

ISBN:1595933646

DOI:10.1145/1179352

Conference Chair:
John Finnegan
Purdue University
,
Program Chair:
Julie Dorsey
Yale University

Copyright © 2006 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Published: 01 July 2006

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SIGGRAPH '06 Paper Acceptance Rate 86 of 474 submissions, 18%;

Overall Acceptance Rate 1,822 of 8,601 submissions, 21%

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Cited By

Gross BAtzberger P(2018)Spectral Numerical Exterior Calculus Methods for Differential Equations on Radial ManifoldsJournal of Scientific Computing10.1007/s10915-017-0617-276:1(145-165)Online publication date: 1-Jul-2018
https://dl.acm.org/doi/10.1007/s10915-017-0617-2
Zhang E(2012)Tensor Field Design: Algorithms and ApplicationsNew Developments in the Visualization and Processing of Tensor Fields10.1007/978-3-642-27343-8_6(111-133)Online publication date: 9-Jul-2012
https://doi.org/10.1007/978-3-642-27343-8_6
Ray NVallet BLi WLévy B(2008)N-symmetry direction field designACM Transactions on Graphics10.1145/1356682.135668327:2(1-13)Online publication date: 8-May-2008
https://dl.acm.org/doi/10.1145/1356682.1356683
Wang KWeiwei Tong YDesbrun MSchröder P(2006)Edge subdivision schemes and the construction of smooth vector fieldsACM Transactions on Graphics10.1145/1141911.114199125:3(1041-1048)Online publication date: 1-Jul-2006
https://dl.acm.org/doi/10.1145/1141911.1141991

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