Surface Conformal Mapping Koebe's Iterative Method

Conformal mapping plays an important role in geometric and physical mod-eling. Most existing techniques can only handle the surface with simple topolo-gies. In this work, we propose a novel method, which can handle surfaces with more complicated topologies.According to the Uniformization theorem, all closed metric surfaces can be conformally mapped to one of the three canonical spaces, the sphere, the plane and the hyperbolic disk. In theory, the uniformization theorem for surfaces with boundaries holds, which claims that all compact metric surfaces with bound-aries can be conformally mapped to circle domains on Riemann surfaces. This work proposes the computational methods for open surface uniformation, the convergence rate is estimated and proved.For genus 0 compact metric surfaces with boundaries, we can apply the gen-eralized Koebe's method. Conventional Koebe's algorithm only handles planar regions, generalized Koebe's method can handle general metric surfaces. Fur-thermore, the generalized Keobe's method converges quadratically faster than that of the conventional method. We also give the proof for the converge order of the generalized Koebe's method.For high genus surface with multiply boundary components, we combine dis-crete surface Ricci flow and Koebe's iteration to compute the canonical conformal mapping.For a genus one surface with multiple boundary components, we apply dis-crete surface Ricci flow, to map its universal covering space onto the Euclidean plane R2, such that all boundary components are mapped to Euclidean circles. We estimate the convergence rate and give detailed proof.For a high genus (g>1) surface with multiple boundary components, we combine discrete surface Ricci flow with Koebe's iteration to map its universal covering space onto the hyperbolic plane H2, such that all boundary components are mapped to hyperbolic circles.Experimental results show the method is general, stable and practical. The method has also been applied for surface matching and shape signature.