Some Questions Of Partial Differential Equation Method In Surface Modeling

It is a new method to generate surfaces with partial differential equation(PDE). The method was initially proposed and was leaded into Computer Aided Design and Computer Graphics by M.I.J Bloor and J. Wilson, in the end of 1980s , who are from leeds university UK. The main idea of PDE method is regarding blending surface as boundary problem of partial differential equation and the solution of equation is what we need.The PDE method was different from the conventional methods. The conventional methods , such as Coons surface modeling method, Bezier method, B-spline method and NURBS method, modified the shape of surface by changing control points , while the PDE method adjusts proper coefficients and boundary conditions and the right terms on the right of partial differential equation. The surface generated with PDE method is smooth, and it is easy to solve partial differential equation for there are a lot of arithmetic methods to solveit.This paper collects some classical algorithms of surface reconstruction and describes the principles of the PDE method for surfaces modeling. Then the author describes an algorithm of surface reconstruction based on the PDE method.The technique of surface reconstruction from unorganized data is very important both theoretically and practically. The data set might consist of points, curves and/or surface patches. The first step of the algorithms is to compartmentalize the region of unorganized data to be triangular meshes, then confirm the boundary conditions of each domain. We can choose the type of elliptic PDE that satisfied all these conditions, the solution of elliptic PDE is what we need. The critical point, in the whole process, is to confirm the boundary conditions of each domain(we can use the difference method and the finitude element method to solve equation). The author adopt cubic—Bezier curve to construct boundary, which avoids producing singular point when adopting quadratic Bezier curve, and create the cross-boundary derivative vector on boundary with the normal vector.Finally, the author suggests the future work to do in related research fields.