Ball surfaces play a crucial role in modeling 3D objects with varying thickness. In this paper, energy functionals for surface design are extended to Ball surfaces and the variational problem of finding the energy-minimizing Ball surface within the constraints imposed by the controls is investigated. Besides, owing to the excellent properties of Bernstein bases, we propose a computationally inexpensive algorithm for the construction of energy-minimizing Ball Bézier surfaces. Finally, an efficient design tool is provided for the construction of C^r continuous energy-minimizing blend surface from two disjoint Ball surfaces. The feasibility of the method is verified by several examples. By adjusting the weighted coefficients, different energy functionals are defined and thus Ball surfaces with different shapes and thickness can be obtained, achieving the deformable modeling of Ball surfaces.

球面在不同厚度的 3D 对象建模中起着至关重要的作用。在本文中，用于表面设计的能量泛函扩展到球面，并研究了在控制施加的约束内寻找能量最小化球面的变分问题。此外，由于 Bernstein 基的优异特性，我们提出了一种计算成本低的算法，用于构建能量最小化的 Ball Bézier 曲面。最后，为C ^r的构建提供了有效的设计工具来自两个不相交的球面的连续能量最小化混合面。通过几个实例验证了该方法的可行性。通过调整加权系数，定义不同的能量函数，从而得到不同形状和厚度的球面，实现球面的可变形建模。