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Canonical conformal mapping for high genus surfaces with boundaries

Authors
Journal
Computers & Graphics
0097-8493
Publisher
Elsevier
Volume
36
Issue
5
Identifiers
DOI: 10.1016/j.cag.2012.03.006
Keywords
  • Uniformization
  • Ricci Flow
  • Koebe'S Iteration
  • Conformal Mapping
  • High Genus Surface With Boundaries
  • Zero Holonomy Condition
Disciplines
  • Engineering
  • Mathematics
  • Medicine

Abstract

Abstract Conformal mapping plays an important role in Computer Graphics and Shape Modeling. According to Poincaré's uniformization theorem, all closed metric surfaces can be conformally mapped to one of the three canonical spaces, the sphere, the plane or the hyperbolic disk. This work generalizes the uniformization from closed high genus surfaces to high genus surfaces with boundaries, to map them to the canonical spaces with circular holes. The method combines discrete surface Ricci flow and Koebe's iteration with zero holonomy condition. Theoretic proof for the convergence is given. Experimental results show that the method is general, stable and practical. It is fundamental and has great potential to geometric analysis in various fields of engineering and medicine.

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