Efficient and robust discrete conformal equivalence with boundary

Marcel Campen, Ryan Capouellez, Hanxiao Shen, Leyi Zhu, Daniele Panozzo, Denis Zorin

Research output: Contribution to journalArticlepeer-review


We describe an efficient algorithm to compute a discrete metric with prescribed Gaussian curvature at all interior vertices and prescribed geodesic curvature along the boundary of a mesh. The metric is (discretely) conformally equivalent to the input metric. Its construction is based on theory developed in [Gu et al. 2018b] and [Springborn 2020], relying on results on hyperbolic ideal Delaunay triangulations. Generality is achieved by considering the surface's intrinsic triangulation as a degree of freedom, and particular attention is paid to the proper treatment of surface boundaries. While via a double cover approach the case with boundary can be reduced to the case without boundary quite naturally, the implied symmetry of the setting causes additional challenges related to stable Delaunay-critical configurations that we address explicitly. We furthermore explore the numerical limits of the approach and derive continuous maps from the discrete metrics.

Original languageEnglish (US)
Article number261
JournalACM Transactions on Graphics
Issue number6
StatePublished - Dec 10 2021


  • cone metric
  • conformal map
  • conformal parametrization
  • edge flip
  • intrinsic delaunay
  • intrinsic triangulation

ASJC Scopus subject areas

  • Computer Graphics and Computer-Aided Design


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