Slit Map : Conformal Parameterization for Multiply Connected Surfaces Xiaotian Yin 1 , Junfei Dai 2 , Shing-Tung Yau 3 , and Xianfeng Gu 1 1 Center for Visual Computing State University of New York at Stony Brook 2 Center of Mathematics Science Zhejiang University 3 Mathematics Department Harvard University Abstract. Surface parameterization is a fundamental tool in geometric modeling and processing. Most existing methods deal with simply con- nected 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 holo- morphic one-form method that requires double covering, it is more effi- cient and has better control over singularities. Compared to the existing Ricci flow 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 accu- racy 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 im- proves the packing efficiency for texture mapping. The positions of the slits are completely determined by the surface geometry, which can be treated as the finger 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. Key words: conformal parameterization, slits, holomorphic one-form 1 Introduction Surface conformal parameterization is a fundamental tool in geometric modeling and processing. It is an essential technique for many applications, such as texture