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.

翻译：曲面参数化在许多科学与工程问题中扮演着基础性角色。特别地，由于零亏格封闭曲面在拓扑上等价于球面，过去几十年已发展出多种球面参数化方法。然而在实际应用中，将零亏格封闭曲面映射到球面可能因几何差异导致较大形变。本研究提出一个新框架，用于计算零亏格封闭曲面的椭球共形与拟共形参数化，其目标参数域为椭球而非球面。通过将简单共形变换与不同类型的拟共形映射相结合，可轻松实现多种椭球参数化，其双射性由拟共形理论保障。数值实验验证了所提框架的有效性。