Discrete Conformal Mappings via Circle Patterns
Please always quote using this URN:urn:nbn:de:0296-matheon-3027
- We introduce a novel method for the construction of discrete conformal mappings from surface meshes of arbitrary topology to the plane. Our approach is based on circle patterns, i.e., arrangements of circles—one for each face—with prescribed intersection angles. Given these angles the circle radii follow as the unique minimizer of a convex energy. The method supports very flexible boundary conditions ranging from free boundaries to control of the boundary shape via prescribed curvatures. Closed meshes of genus zero can be parameterized over the sphere. To parameterize higher genus meshes we introduce cone singularities at designated vertices. The parameter domain is then a piecewise Euclidean surface. Cone singularities can also help to reduce the often very large area distortion of global conformal maps to moderate levels. Our method involves two optimization problems: a quadratic program and the unconstrained minimization of the circle pattern energy. The latter is a convex function of logarithmic radius variables with simple explicit expressions for gradient and Hessian. We demonstrate the versatility and performance of our algorithm with a variety of examples.
Author: | Liliya Kharevych, Boris A. Springborn, Peter Schröder |
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URN: | urn:nbn:de:0296-matheon-3027 |
Referee: | Günter M. Ziegler |
Document Type: | Preprint, Research Center Matheon |
Language: | English |
Date of first Publication: | 2006/03/14 |
Release Date: | 2006/03/14 |
Tag: | |
Institute: | Technische Universität Berlin |
Zuse Institute Berlin (ZIB) | |
Preprint Number: | 313 |