Here, we extend the finite distortion problem from bounded domains in ℝ2 to closed genus-zero surfaces in ℝ3 by a stereographic projection. Then, we derive a theoretical foundation for spherical equiareal parameterization between a simply connected closed surface M and a unit sphere 𝕊2 by minimizing the total area distortion energy on ̅ℂ. After the minimizer of the total area distortion energy is determined, it is combined with an initial conformal map to determine the equiareal map between the extended planes. From the inverse stereographic projection, we derive the equiareal map between M and 𝕊2. The total area distortion energy is rewritten into the sum of Dirichlet energies associated with the southern and northern hemispheres and is decreased by alternatingly solving the corresponding Laplacian equations. Based on this foundational theory, we develop a modified stretch energy minimization function for the computation of spherical equiareal parameterization between M and 𝕊2. In addition, under relatively mild conditions, we verify that our proposed algorithm has asymptotic R-linear convergence or forms a quasi-periodic solution. Numerical experiments on various benchmarks validate that the assumptions for convergence always hold and indicate the efficiency, reliability, and robustness of the developed modified stretch energy minimization function.

中文翻译：

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球面等面参数化拉伸能量最小化的基本理论和R线性收敛

在这里，我们从 ℝ 中的有界域扩展有限失真问题2到 ℝ 中的闭亏格零曲面3通过立体投影。然后，我们推导了简单连通封闭曲面之间的球面等面积参数化的理论基础中号和一个单位球体 𝕊2通过最小化 ̅ℂ 上的总面积畸变能量。在确定总面积畸变能量的最小化值之后，将其与初始共角图结合以确定扩展平面之间的等面积图。从逆立体投影中，我们推导出之间的等面积图中号和𝕊2。总面积畸变能量被重写为与南半球和北半球相关的狄利克雷能量之和，并通过交替求解相应的拉普拉斯方程来减少。基于这一基础理论，我们开发了一种改进的拉伸能量最小化函数，用于计算之间的球面等面积参数化中号和𝕊2。此外，在相对温和的条件下，我们验证了我们提出的算法具有渐近R线性收敛或形成准周期解。各种基准的数值实验验证了收敛假设始终成立，并表明了所开发的改进的拉伸能量最小化函数的效率、可靠性和鲁棒性。