Communications in Information and Systems

Volume 2 (2002)

Number 2

Computing conformal structures of surfaces

Pages: 121 – 146



Xianfeng Gu (Division of Engineering and Applied Science, Harvard University, Cambridge, Massachusetts, U.S.A.)

Shing-Tung Yau (Mathematics Department, Harvard University, Cambridge, Massachusetts, U.S.A.)


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.


mesh, conformal structure, texture mapping, holomorphic forms, harmonic forms, period matrix

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