Generating Bézier Surfaces Based On 8th-order PDE

In this paper, we make an in-depth study of the surface-design problem in Computer Aided Geometric Design(CAGD), and propose a Bezier-surface designing method based on a general 8-th order PDE. In the introduction, we take a brief review of several types of surface modeling method in CAGD, and introduce the development of the PDE surfaces in CAGD, then introduce the development of several special types of PDE surfaces such as the harmonic Bezier surfaces, the biharmonic Bezier surfaces, the tetraharmonic Bezier surfaces. Based on the above researches, we propose a Bezier-surface designing method based on general 8-th order PDE. The main innovative contributions of this paper can be summarized as follows:1.The Spanish scholar Monterde extended the work on constructing biharmonic Bezier surfaces([Monterde,2004b]) to finding polynomial solution surfaces of a general 4-th order PDE([Monterde,2006]), by similar way, we extend his work to finding solution surfaces of a general 8-th order PDE. The former solution surfaces only interpolate the boundary position, but the latter solution surfaces interpolate both the boundary position and the boundary tangent vector, and the surface shape can be adjusted by the eight equation coefficients, therefore our work not only deepens the theory of higher order polynomial solutions of PDE, but also increases the grade in engineering applications, and will meet the requirements of more users.2. We point out that the solution of a 8-th order PDE with any coefficients actually can be seen as the extremal of the corresponding fourth-order quadratic functional, and further rigorously deduce its complete solution.3. We demonstrate that the tetraharmonic Bezier surfaces and the Coons bicubic blending surfaces which have significant application value in CAD are the special cases of our results, so our work specially boosts the harmonic surface and minimal surface designing which are widely used in modern architecture and engineering modeling.