A Unified Approach To Geometric Modeling Of Curves And Surfaces

A unified approach to geometric modeling of curves and surfaces is given. Both a vector-valued fourth and sixth order partial differential equations (PDEs) of motion are proposed. The fourth order PDE covers all existing PDEs used for surface modeling, and the sixth order PDE considers the curvature effect on curves and surfaces. In order to apply these PDEs to create curves and surfaces in real time, we have presented a composite power series method which guarantees the exact satisfaction of boundary conditions, and represents curves and surfaces with analytical mathematical formulae. We have examined the accuracy and efficiency of the proposed method, and employed it to a number of applications of static and dynamic modeling of curves and surfaces, including free-form surface generation and surface blending. It is found that this method has similar computational accuracy and efficiency to the corresponding closed form solution method, and creates curves and surfaces far more efficiently and accurately than numerical methods. In addition, it can deal with complicated shape modeling problems.