  Conformal mapping, a classical topic in complex analysis and differential
geometry, has become a subject of great interest in the area of surface
parameterization in recent decades with various applications in science and
engineering. However, most of the existing conformal parameterization
algorithms only focus on simply-connected surfaces and cannot be directly
applied to surfaces with holes. In this work, we propose two novel algorithms
for computing the conformal parameterization of multiply-connected surfaces. We
first develop an efficient method for conformally parameterizing an open
surface with one hole to an annulus on the plane. Based on this method, we then
develop an efficient method for conformally parameterizing an open surface with
$k$ holes onto a unit disk with $k$ circular holes. The conformality and
bijectivity of the mappings are ensured by quasi-conformal theory. Numerical
experiments and applications are presented to demonstrate the effectiveness of
the proposed methods.
