Optimal Global Conformal Surface Parameterizatio
Miao Jin, Yalin Wang, Shing-Tung Yau, and Xianfeng Gu
All orientable metric surfaces are Riemann surfaces and admit
global conformal parameterizations. Riemann surface structure is a
fundamental structure and governs many natural physical phenomena,
such as heat diffusion and electro-magnetic fields on the surface.
A good parameterization is crucial for simulation and visualization.
This paper provides an explicit method for finding optimal
global conformal parameterizations of arbitrary surfaces. It relies
on certain holomorphic differential forms and conformal mappings
from differential geometry and Riemann surface theories. Algorithms
are developed to modify topology, locate zero points, and
determine cohomology types of differential forms. The implementation
is based on a finite dimensional optimization method. The
optimal parameterization is intrinsic to the geometry, preserves angular
structure, and can play an important role in various applications
including texture mapping, remeshing, morphing and simulation.
The method is demonstrated by visualizing the Riemann
surface structure of real surfaces represented as triangle meshes.
Figures (click on each for a larger version):