Adjusting the energy of Ball surfaces by modifying unfixed control balls

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.