## Discrete One-Forms on Meshes and Applications to 3D Mesh Parameterization (2006)

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Venue: | Journal of CAGD |

Citations: | 25 - 1 self |

### BibTeX

@ARTICLE{Gortler06discreteone-forms,

author = {Steven J. Gortler and Craig Gotsman and Dylan Thurston},

title = {Discrete One-Forms on Meshes and Applications to 3D Mesh Parameterization},

journal = {Journal of CAGD},

year = {2006},

pages = {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.