Metric Optimzation in Penner Coordinates
Abstract
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.
Paper
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Results
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.
BibTex
@article{capouellez:2023:penner,
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 = {https://doi.org/10.1145/3618394},
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}
}