Fair Morse functions for extracting the topological structure of a surface mesh

Xinlai Ni, Michael Garland, John C. Hart

Research output: Contribution to journalConference articlepeer-review


Morse theory reveals the topological structure of a shape based on the critical points of a real function over the shape. A poor choice of this real function can lead to a complex configuration of an unnecessarily high number of critical points. This paper solves a relaxed form of Laplace's equation to find a "fair" Morse function with a user-controlled number and configuration of critical points. When the number is minimal, the resulting Morse complex cuts the shape into a disk. Specifying additional critical points at surface features yields a base domain that better represents the geometry and shares the same topology as the original mesh, and can also cluster a mesh into approximately developable patches. We make Morse theory on meshes more robust with teflon saddles and flat edge collapses, and devise a new "intermediate value propagation" multigrid solver for finding fair Morse functions that runs in provably linear time.

Original languageEnglish (US)
Pages (from-to)613-622
Number of pages10
JournalACM Transactions on Graphics
Issue number3
StatePublished - 2004
EventACM Transactions on Graphics - Proceedings of ACM SIGGRAPH 2004 -
Duration: Aug 9 2004Aug 12 2004


  • Atlas generation
  • Computational topology
  • Morse theory
  • Surface parameterization
  • Texture mapping

ASJC Scopus subject areas

  • Computer Graphics and Computer-Aided Design


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