  Surface parameterization plays a fundamental role in many science and
engineering problems. In particular, as genus-0 closed surfaces are
topologically equivalent to a sphere, many spherical parameterization methods
have been developed over the past few decades. However, in practice, mapping a
genus-0 closed surface onto a sphere may result in a large distortion due to
their geometric difference. In this work, we propose a new framework for
computing ellipsoidal conformal and quasi-conformal parameterizations of
genus-0 closed surfaces, in which the target parameter domain is an ellipsoid
instead of a sphere. By combining simple conformal transformations with
different types of quasi-conformal mappings, we can easily achieve a large
variety of ellipsoidal parameterizations with their bijectivity guaranteed by
quasi-conformal theory. Numerical experiments are presented to demonstrate the
effectiveness of the proposed framework.
