Metric Optimzation in Penner Coordinates

Ryan Capouellez1, Denis Zorin1
1New York University


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.




Examples of conformally and isometrically parameterized models: closed, with boundary, cut with fixed boundary angles. For each model, we visualize local scale distortion and distribution of per-vertex scale factors.


  author = {Capouellez, Ryan and Zorin, Denis},
  title = {Metric Optimization in Penner Coordinates},
  year = {2023},
  issue_date = {December 2023},
  publisher = {Association for Computing Machinery},
  address = {New York, NY, USA},
  volume = {42},
  number = {6},
  issn = {0730-0301},
  url = {},
  doi = {10.1145/3618394},
  journal = {ACM Trans. Graph.},
  month = {dec},
  articleno = {234},
  numpages = {19},
  keywords = {cone metrics, conformal mapping, discrete metrics, intrinsic triangulation, parametrization, penner coordinates}