### Abstract

Surface parameterization refers to the process of mapping the surface to canonical planar domains, which plays crucial roles in texture mapping and shape analysis purposes. Most existing techniques focus on simply connected surfaces. It is a challenging problem for multiply connected genus zero surfaces. This work generalizes conventional Koebe's method for multiply connected planar domains. According to Koebe's uniformization theory, all genus zero multiply connected surfaces can be mapped to a planar disk with multiply circular holes. Furthermore, this kind of mappings are angle preserving and differ by Möbius transformations. We introduce a practical algorithm to explicitly construct such a circular conformal mapping. Our algorithm pipeline is as follows: suppose the input surface has n boundaries, first we choose 2 boundaries, and fill the other n - 2 boundaries to get a topological annulus; then we apply discrete Yamabe flow method to conformally map the topological annulus to a planar annulus; then we remove the filled patches to get a planar multiply connected domain. We repeat this step for the planar domain iteratively. The two chosen boundaries differ from step to step. The iterative construction leads to the desired conformal mapping, such that all the boundaries are mapped to circles. In theory, this method converges quadratically faster than conventional Koebe's method. We give theoretic proof and estimation for the converging rate. In practice, it is much more robust and efficient than conventional non-linear methods based on curvature flow. Experimental results demonstrate the robustness and efficiency of the method.

Original language | English (US) |
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Title of host publication | Proceedings - SPM 2009 |

Subtitle of host publication | SIAM/ACM Joint Conference on Geometric and Physical Modeling |

Pages | 89-100 |

Number of pages | 12 |

DOIs | |

State | Published - Nov 9 2009 |

Event | SPM 2009: SIAM/ACM Joint Conference on Geometric and Physical Modeling - San Francisco, CA, United States Duration: Oct 5 2009 → Oct 8 2009 |

### Publication series

Name | Proceedings - SPM 2009: SIAM/ACM Joint Conference on Geometric and Physical Modeling |
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### Other

Other | SPM 2009: SIAM/ACM Joint Conference on Geometric and Physical Modeling |
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Country | United States |

City | San Francisco, CA |

Period | 10/5/09 → 10/8/09 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Computational Theory and Mathematics
- Computer Science Applications
- Computer Vision and Pattern Recognition
- Mathematics(all)

### Keywords

- Circular
- Conformal
- Differential form
- Holomorphic
- Multiply connected domain
- Uniformization

### Cite this

*Proceedings - SPM 2009: SIAM/ACM Joint Conference on Geometric and Physical Modeling*(pp. 89-100). [1629267] (Proceedings - SPM 2009: SIAM/ACM Joint Conference on Geometric and Physical Modeling). https://doi.org/10.1145/1629255.1629267