Surface parameterization is a fundamental concept in fields such as differential geometry and computer graphics. It involves mapping a surface in three-dimensional space onto a two-dimensional parameter space. This process allows for the systematic representation and manipulation of surfaces of complicated shapes by simplifying them into a manageable planar domain. In this paper, we propose a new iterative algorithm for computing the parameterization of simply connected open surfaces that achieves an optimal balance between angle and area distortions. We rigorously prove that the iteration in our algorithm converges globally, and numerical results demonstrate that the resulting mappings are bijective and effectively balance angular and area accuracy across various triangular meshes. Additionally, we present the practical usefulness of the proposed algorithm by applying it to represent surfaces as geometry images.

翻译：曲面参数化是微分几何与计算机图形学等领域的基础概念，其核心在于将三维空间中的曲面映射至二维参数空间。这一过程通过将复杂形状的曲面简化为可处理的平面域，实现了曲面的系统化表示与操作。本文提出一种新的迭代算法，用于计算单连通开放曲面的参数化，该算法能够在角度失真与面积失真之间达到最优平衡。我们严格证明了算法中迭代过程的全局收敛性，数值结果表明所得映射是双射的，并能在不同三角网格上有效平衡角度与面积的精度。此外，通过将所提算法应用于将曲面表示为几何图像，我们展示了其实际应用价值。