Blending surface generation using a fast and accurate analytical solution of a fourth-order PDE with three shape control parameters

Abstract

In this paper, we propose to use a fourth-order partial differential equation (PDE) to solve a class of surface-blending problems. This equation has three vector-valued shape control parameters. It incorporates all the previously published forms of fourth-order PDEs for surface blending and can generate a larger class of blending surfaces than existing equations. To apply the proposed PDE to the solution of various blending problems, we have developed a fast and accurate resolution method. Our method modifies Navier’s solution for the elastic bending deformation of thin plates by making it satisfy the boundary conditions exactly. A comparison between our method, the closed-form solution method, and other existing analytical methods indicates that the developed method is able to generate blending surfaces almost as quickly and accurately as the closed-form solution method, far more efficiently and accurately than the numerical methods and other existing analytical methods. Having investigated the effects that the vector-valued shape parameters and the force function of the proposed equation have on the blending surface, we have found that they have a significant influence on its shape. They provide flexible user handles that surface designers can use to adjust the blending surface to acquire the desired shape. The developed method was employed in the investigation of surface-blending problems where the primary surfaces were expressed in parametric, implicit, and explicit forms.