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Published April 2006 | public
Journal Article

Discrete Conformal Mappings via Circle Patterns

Abstract

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, that is, 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.

Additional Information

© 2006 ACM. Received July 2005; revised September 2005; accepted October 2005. This work was supported in part by NSF (DMS-0220905, DMS-0138458, ACI-0219979), DFG Research Center MATHEON "Mathematics for key technologies", DOE (W-7405-ENG-48/B341492), Center for the Mathematics of Information, Alias, and Pixar. Special thanks to Alexander Bobenko, Mathieu Desbrun, Ilja Friedel, Nathan Litke, and Cici Koenig.

Additional details

Created:
August 19, 2023
Modified:
October 23, 2023