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.