Computing Conformal Structures of Surfaces
Xianfeng Gu and Shing-Tung Yau
Communications in Information and Systems 2(2), 2002, 121-146
This paper solves the problem of computing conformal structures of
general 2-manifolds represented as triangular meshes. We approximate the De Rham
cohomology by simplicial cohomology and represent the Laplace-Beltrami operator,
the Hodge star operator by linear systems. A basis of holomorphic one-forms is
constructed explicitly. We then obtain a period matrix by integrating
holomorphic differentials along a homology basis. We also study the global
conformal mappings between genus zero surfaces and spheres, and between general
surfaces and planes. Our method of computing conformal structures can be applied
to tackle fundamental problems in computer aid geometry design and computer
graphics, such as geometry classification and identification, and surface global
parametrization.