Generation of interpolation surfaces with the least strain energy using dynamic-programming

Construction of a surface is usually executed from the viewpoint of either interpolation or approximation (in other words, conceptual design). From the viewpoint of interpolation, smooth interpolation among given points that is compatible with physical phenomena is often essential. A surface with the minimum strain energy is known as a smooth surface satisfying those requirements. Some methods for construction of an interpolation surface use the strain energy approximated in a suitable way and generate a surface for which the approximated strain energy is minimum. Nevertheless, generated surfaces sometimes contain ‘wiggles’ or ‘bumps’ for data which implies large gradients. The purpose of this paper is to initiate the method for generating the surface with little ‘wiggles’ or ‘bumps’ for data which implies large gradients. For this purpose, we first adopt the minimization of the meansquare curvature in the x- and y-directions in a surface as a criterion to estimate the propriety of the interpolation. Next, we derive the optimum equation satisfied by the surface with the minimum meansquare curvature by representing the surface in the form of the C-1 Coons patch and then applying dynamic programming to the minimization problem. Finally, the solution to the optimality equation is obtained by a numerical method, and the surface with the minimum meansquare curvature is generated.