Surface Modeling Method Based On The Helmholtz Equation

With the spreading application of CAD/CAM, people place more and more high acquirements on CAGD. The traditional NURBS based surfaces modeling can't satisfy the needs to construct the complicated surfaces and generate high quality shape. This is a major challenge facing system on CAGD and has given rise to a number of alternative approaches: the wavelets-based surface modeling, the PDE surface modeling, the energy-based surface modeling and so on.In order to pursue the trends of advanced research, this paper is about the study of surface modeling approach which based on the Helmholtz equations. Helmholtz equations can be introduced into the design of surface modeling. For the purpose of getting more free parameters in surface modeling design, to expend the coefficients of Helmholtz equations, given a class of partial differential equations which contain with the shape of the three control parameters, being called bi-Helmholtz equation in this article. This pepper also focused on second-order and fourth-order of the bi-Helmholtz equations to application in the transition area structure and shape control and free-form design. It's studied the shape of the control parameters on the shape of the surface.To improve the design capabilities of the interactive surface modeling, this paper get a six vector shape function of the bi-Helmholtz equations to discuss the shape of the surface influenced by the shape control parameters and its power source function, It's modifying the boundary conditions and Arrow cut boundary to achieve interactive. As the vector shape function is a continuous function based on surface parameters u,v of self variables. It's can be very throughout the parameters domain. So it's has greatly increased the surface construct abilities for use of partial differential equations (PDE) method. It's also provides greater flexibility for interactive Surface design, and enhanced surfaces control. In addition, this article discussed the numerical methods which structure a PDE surface modeling, that is, the point spectral-allocation method, detailed in the differential spectral matrix calculation and its application in PDE surface modeling design.