Slit Map: Conformal Parameterization for Multiply Connected Surfaces

Surface parameterization is a fundamental tool in geometric modeling and processing. Different to most existing methods, this work introduces a

linear

method, called

slit map

, to conformally parameterize multiply connected disk-like surfaces with

canonical

domains, namely

circular slit domain

(annulus with concentric circular slits) or

parallel slit domain

(rectangle with parallel slits) equivalently. The construction is based on holomorphic one-forms, which is intrinsic, automatic and efficient. Slit map owns many merits over existing methods. The regularity of the image boundaries simplifies many geometric process, such as surface matching and etc; the full utilization of the parallel slit texture domain improves the packing efficiency for texture mapping; the positions of the slits are completely determined by the surface geometry and can be treated as the finger prints for surface classification. In the paper, both the underlying theory and the algorithm pipeline are explained. Preliminary experimental results are shown for several application purposes.