A novel method for the construction of discrete conformal mappings from (regions of) embedded meshes to the plane is presented here. The 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 has two principal advantages over earlier approaches based on discrete harmonic mappings:
(1) It supports very flexible boundary conditions ranging from natural boundaries to control of the boundary shape via prescribed curvatures.
(2) The solution is based on a convex energy as a function of logarithmic radius variables with simple explicit expressions for gradients and Hessians, greatly facilitating robust and efficient numerical treatment.
The versatility and performance of our algorithm is demonstrated with a variety of examples.
Liliya Kharevych, Boris Springborn, and Peter Schröder.
Discrete Conformal Mappings via Circle Patterns.
ACM Trans. Graph., 2006.