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Published July 2006 | public
Journal Article

Edge Subdivision Schemes and the Construction of Smooth Vector Fields


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.

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.

Additional details

August 22, 2023
October 23, 2023