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：表面参数化在众多科学与工程问题中具有基础性作用。由于零亏格封闭曲面在拓扑上等价于球面，过去几十年间涌现出大量球面参数化方法。然而在实际应用中，由于几何差异，将零亏格封闭曲面映射至球面可能导致较大形变。本文提出一种新型计算框架，用于实现零亏格封闭曲面到椭球面的共形与拟共形参数化——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，可便捷获得丰富多样的椭球参数化结果，且其双射性可由拟共形理论予以保证。数值实验验证了所提框架的有效性。