Computing conformal structure of surfaces

Xianfeng Gu Shing-Tung Yau

Complex Variables and Complex Analysis mathscidoc:1912.43484

arXiv preprint cs/0212043, 2002.12
This paper solves the problem of computing conformal structures of general 2-manifolds represented as triangle meshes. We compute conformal structures in the following way: first compute homology bases from simplicial complex structures, then construct dual cohomology bases and diffuse them to harmonic 1-forms. Next, we construct bases of holomorphic differentials. We then obtain period matrices by integrating holomorphic differentials along homology bases. We also study the global conformal mapping between genus zero surfaces and spheres, and between general meshes and planes. Our method of computing conformal structures can be applied to tackle fundamental problems in computer aid design and computer graphics, such as geometry classification and identification, and surface global parametrization.
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@inproceedings{xianfeng2002computing,
  title={Computing conformal structure of surfaces},
  author={Xianfeng Gu, and Shing-Tung Yau},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224203535496761048},
  booktitle={arXiv preprint cs/0212043},
  year={2002},
}
Xianfeng Gu, and Shing-Tung Yau. Computing conformal structure of surfaces. 2002. In arXiv preprint cs/0212043. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224203535496761048.
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