Discrete one-forms on meshes and applications to 3D mesh parameterization

Steven J. Gortler, Craig Gotsman, Dylan Thurston

Research output: Contribution to journalArticlepeer-review

99 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)83-112
Number of pages30
JournalComputer Aided Geometric Design
Issue number2
StatePublished - Feb 2006
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Modeling and Simulation
  • Automotive Engineering
  • Aerospace Engineering
  • Computer Graphics and Computer-Aided Design


  • Computer graphics
  • Embedding
  • Manifold mesh
  • One-form
  • Parameterization


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