Toward Computing Extremal Quasiconformal Maps via Discrete Harmonic Measured Foliations

Abstract

Conformal maps, which preserve angles, are widely used in computer graphics for applications such as parameterization. However, conformal maps are usually unavailable between multiply-connected domains or when extra constraints are imposed. Extremal quasiconformal maps naturally generalize conformal maps and are available under a larger class of constraints. Teichmüller’s theorem guarantees the existence of such maps and describes them in terms of holomorphic quadratic differentials. This enables computation of extremal maps by first computing holomorphic quadratic differentials on the domain and range. In this thesis, we take the first step toward that project. We define a new type of object—a discrete measured foliation—and prove some results analogous to those for smooth measured foliations. In particular, measured foliations form topological equivalence classes called Whitehead classes, and each Whitehead class contains a unique "harmonic" representative corresponding to a holomorphic quadratic differential. Finally, we develop an algorithm for evolving any discrete foliation to the harmonic representative in its Whitehead class.