Discrete One-forms on Meshes and Applications to 3D mesh Parameterization

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Discrete One-forms on Meshes and Applications to 3D mesh Parameterization

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Title: Discrete One-forms on Meshes and Applications to 3D mesh Parameterization
Author: Thurston, Dylan; Gortler, Steven; Gotsman, Craig

Note: Order does not necessarily reflect citation order of authors.

Citation: Gortler, Steven. J., Craig Gotsman, and Dylan Thurston. 2006. Discrete one-forms on meshes and applications to 3D mesh parameterization. Computer Aided Geometric Design 23(2): 83-112.
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Abstract: We describe how some simple properties of discrete one-forms directly relate to some old and new results concerning the parameterization of 3D mesh data. Our first result is an easy proof of Tutte's celebrated “spring-embedding” theorem for planar graphs, which is widely used for parameterizing meshes with the topology of a disk as a planar embedding with a convex boundary. Our second result generalizes the first, dealing with the case where the mesh contains multiple boundaries, which are free to be non-convex in the embedding. We characterize when it is still possible to achieve an embedding, despite these boundaries being non-convex. The third result is an analogous embedding theorem for meshes with genus 1 (topologically equivalent to the torus). Applications of these results to the parameterization of meshes with disk and toroidal topologies are demonstrated. Extensions to higher genus meshes are discussed.
Published Version: http://dx.doi.org/10.1016/j.cagd.2005.05.002
Terms of Use: This article is made available under the terms and conditions applicable to Other Posted Material, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#LAA
Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:2634288
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