Conformal Spherical Parametrization for High Genus Surfaces

Abstract

Surface parameterization establishes bijective maps from a surface onto atopologically equivalent standard domain. It is well known that the sphericalparameterization is limited to genus-zero surfaces. In this work, we design anew parameter domain, two-layered sphere, and present a framework for mappinghigh genus surfaces onto sphere. This setup allows us to trans- fer the existingapplications based on general spherical parameterization to the field of highgenus surfaces, such as remeshing, consistent parameterization, shape analysis,and so on. Our method is based on Riemann surface theory. We constructmeromorphic functions on sur- faces: for genus one surfaces, we applyWeierstrass {$P$-functions;} for high genus surfaces, we compute the quotientbetween two holomorphic one-forms. Our method of spherical parameterization istheoretically sound and practically efficient. It makes the subsequentapplications on high genus surfaces very promising.