Metric Optimization in Penner Coordinates

TL;DR

The paper introduces Penner coordinates as a global, connectivity-agnostic parameterization of the space of flat cone metrics on a fixed vertex set, enabling metric optimization and interpolation beyond conformal maps. It formulates optimization problems in this space, derives gradient calculations, and develops both unconstrained (in shear coordinates) and constrained (coordinate-projected) approaches to enforce angle constraints while allowing connectivity changes. By linking Penner coordinates to decorated ideal hyperbolic metrics and using Ptolemy transitions, it constructs continuous maps between metric spaces and provides a practical framework for metric interpolation, distortion control, and refinement-aware mappings. Experiments on challenging datasets demonstrate substantial distortion reductions under angle constraints and highlight the method’s robustness and flexibility, while acknowledging computational speed as an area for future improvement.

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.

Figures (20)

• Figure 1: Intrinsic edge flip: two adjacent triangles are unfolded to the plane, and then a standard flip is performed; the flipped edge corresponds to a broken line on the original geometry.
• Figure 2: A schematic illustration of transitions between different Penner cells.
• Figure 3: Example of a Penner cell partition of the space of cone metrics with 3 vertices and 3 edges. It has four cells, corresponding to mesh connectivities with vertex degrees (2,2,2), and 3 versions with degrees (1,1,4), corresponding to three possible flips. The cells are shown in logarithmic coordinates $\ln a$, $\ln b$, in the plane with equation $\ln a + \ln b+ \ln e = 0$. We show two triangles of each configuration laid out in the plane after cuts along two edges, and on a sphere as curved triangles, to illustrate the connectivity of the edge graph more explicitly.
• Figure 4: Log length energy landscapes for shear optimization of a tetrahedron with different angle constraints: regular tetrahedron has all vertex angles equal, and "flat" tetrahedron has $2\pi$ total angle at one vertex and equal $2\pi/3$ at the remaining ones. Note that the second landscape is considerably less smooth.
• Figure 5: Two-triangle chart for the ideal hyperbolic metric construction.

Theorems & Definitions (18)

• definition 1
• definition 2
• Proposition 1
• definition 3
• Proposition 2
• proof
• definition 4
• Proposition 3
• definition 5
• Proposition 4