  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.
