Computation of Quasi-Conformal Surface Maps Using Discrete Beltrami Flow

Abstract

The manipulation of surface homeomorphisms is an important aspect in three-dimensional modeling and surface processing. Every homeomorphic surface map can be considered as a quasi-conformal map, with its local nonconformal distortion given by its Beltrami differential. As a generalization of conformal maps, quasi-conformal maps are of great interest in mathematical study and real applications. Efficient and accurate computational construction of desirable quasi-conformal maps between general surfaces is crucial. However, in the literature we have reviewed, all existing computational works on construction of quasi-conformal maps to or from a compact domain require global parametrization onto the plane and are difficult to directly apply to maps between arbitrary surfaces. This work fills the gap by proposing to compute quasi-conformal homeomorphisms between arbitrary Riemann surfaces using discrete Beltrami flow, which is a vector field corresponding to the adjustment to the intrinsic Beltrami differential...