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Edge subdivision schemes and the construction of smooth vector fields

Wang, Ke and Weiwei and Tong, Yiying and Desbrun, Mathieu and Schröder, Peter (2006) Edge subdivision schemes and the construction of smooth vector fields. In: SIGGRAPH '06 ACM SIGGRAPH 2006 Papers. ACM , New York, NY, pp. 1041-1048. ISBN 1-59593-364-6. https://resolver.caltech.edu/CaltechAUTHORS:20170104-161620229

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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.


Item Type:Book Section
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1145/1179352.1141991DOIArticle
http://dl.acm.org/citation.cfm?doid=1179352.1141991PublisherArticle
ORCID:
AuthorORCID
Desbrun, Mathieu0000-0003-3424-6079
Schröder, Peter0000-0002-0323-7674
Additional Information:© 2006 ACM. 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.
Funders:
Funding AgencyGrant Number
NSFCCF-0528101
NSFCCR-0133983
NSFDMS-0453145
Department of Energy (DOE)W-7405-ENG-48/B341492
Department of Energy (DOE)DE-FG02-04ER25657
Caltech Center for Mathematics of InformationUNSPECIFIED
nVidiaUNSPECIFIED
AutodeskUNSPECIFIED
Subject Keywords:Subdivision, Discrete Exterior Calculus, Discrete Differential Geometry, vector fields, smooth surface modeling
Classification Code:I.3.5 [ Computer Graphics ] : Computational Geometry and Object Modeling— Curve, surface, solid, and object representations, Physically based modeling, Splines; J.6 [ Computer Applications ] : Computer-Aided Engineering— Computer-aided design (CAD)
Record Number:CaltechAUTHORS:20170104-161620229
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20170104-161620229
Official Citation:Ke Wang, Weiwei, Yiying Tong, Mathieu Desbrun, and Peter Schröder. 2006. Edge subdivision schemes and the construction of smooth vector fields. In ACM SIGGRAPH 2006 Papers (SIGGRAPH '06). ACM, New York, NY, USA, 1041-1048. DOI=http://dx.doi.org/10.1145/1179352.1141991
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:73243
Collection:CaltechAUTHORS
Deposited By: Kristin Buxton
Deposited On:05 Jan 2017 00:30
Last Modified:09 Mar 2020 13:19

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