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

View/ Open
Published Version
https://doi.org/10.1016/j.cagd.2005.05.002Metadata
Show full item recordCitation
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.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.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#LAACitable link to this page
http://nrs.harvard.edu/urn-3:HUL.InstRepos:2634288
Collections
- FAS Scholarly Articles [14543]
Contact administrator regarding this item (to report mistakes or request changes)