Slit Map: Conformal Parameterization for Multiply Connected Surfaces


GMP 2008
Xiaotian Yin, Junfei Dai, Shing-Tung Yau and Xianfeng Gu
Surface parameterization is a fundamental tool in geometric modeling and processing. Most existing methods deal with simply connected disks. This work introduces a novel method to handle multiply connected surfaces based on holomorphic one-forms. The method maps genus zero surfaces with arbitrary number of boundaries to an annulus with concentric circular slits. Any two boundaries can be chosen to map to the inner circle and the outer circle, the other boundaries to slits. Equivalently, the surfaces can be mapped to a rectangle with horizontal slits. Compared to existing linear methods that require surface partition, this method is more intrinsic and automatic. Compared to the existing holomorphic one-form method that requires double covering, it is more e- cient and has better control over singularities. Compared to the existing Ricci ow method, this one is linear and simpler. The proposed method has many merits. The images of boundaries are parallel line segments. This regularity not only helps improve the accuracy for surface matching with boundaries, but also makes quad-remeshing or mesh-spline conversion conversion convenient. The whole rectangle in texture domain is fully occupied without any gap or overlapping; this improves the packing eciency for texture mapping. The positions of the slits are completely determined by the surface geometry, which can be treated as the nger print of the surface to classify surfaces by conformal equivalence. The algorithm is thoroughly explained in detail. Experimental results are demonstrated to show the usefulness of the algorithm for multiply connected domains.