Metric Optimization in Penner Coordinates

Ryan Capouellez, Denis Zorin

Research output: Contribution to journalArticlepeer-review


Many parametrization and mapping-related problems in geometry processing can be viewed as metric optimization problems, i.e., computing a metric minimizing a functional and satisfying a set of constraints, such as flatness. Penner coordinates are global coordinates on the space of metrics on meshes with a fixed vertex set and topology, but varying connectivity, making it homeomorphic to the Euclidean space of dimension equal to the number of edges in the mesh, without any additional constraints imposed. These coordinates play an important role in the theory of discrete conformal maps, enabling recent development of highly robust algorithms with convergence and solution existence guarantees for computing such maps. We demonstrate how Penner coordinates can be used to solve a general class of optimization problems involving metrics, including optimization and interpolation, while retaining the key solution existence guarantees available for discrete conformal maps.

Original languageEnglish (US)
Article number234
JournalACM Transactions on Graphics
Issue number6
StatePublished - Dec 4 2023


  • cone metrics
  • conformal mapping
  • discrete metrics
  • intrinsic triangulation
  • Parametrization
  • Penner coordinates

ASJC Scopus subject areas

  • Computer Graphics and Computer-Aided Design


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