Discrete One-Forms on Meshes and Applications to 3D Mesh Parameterization

Authors

Steven J. Gortler; Craig Gotsman; Dylan Thurston.



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.

BibTex entry

@article { 196, title = {Discrete One-Forms on Meshes and Applications to 3D Mesh Parameterization}, journal = {Computer Aided Geometric Design}, volume = {23}, year = {2006}, month = {02/2006}, pages = {83-112}, publisher = {Elsevier}, chapter = {83}, author = {Steven J. Gortler and Craig Gotsman and Dylan Thurston.} }