构造代数Blending曲面的Gr bner基方法

The Method of Gr bner Basis for Constructing Algebraic Blending Surfaces

利用代数几何中关于理想的 Gr bner基的理论 ,结合 CAGD中的研究方法 ,对代数 Blending曲面做了较为细致的研究 ,给出了用 Gr bner基构造代数 Blending曲面的新方法 .该方法能够求出所有满足要求的代数Blending曲面 ,并能给出其中次数最低的曲面 .文中还讨论了如何利用代数曲面插值、最小平方逼近的方法来选取合适的自由参数 ,以达到对代数 Blending曲面进行形状控制的目的 .最后给出了一个茶壶表面造型示例 。

The study of constructing blending surfaces between given surfaces is one of the important problems in geometric modeling and computer graphics. In the past decades, many authors work on this problem and different methods are proposed to solve the problem. However, these methods generally produce high degree algebraic blending surfaces which are unsatisfactory or the algorithms are not easy to be generalized to blend algebraic surfaces with higher order contact. In this paper, a new method using Grobner basis in algebraic geometry combined with techniques in CAGD for constructing algebraic blending surfaces is introduced. At first a theorem characterized all elements in an ideal which the degree of them is not greater than m using Grobner basis with Graded Lex order is proved. According to this theorem all algebraic blending surfaces which satisfy the given conditions can be obtained, furthermore, the lowest degree blending surfaces also can be found. Based on geometric continuous condition between algebraic surfaces, a general algorithm for constructing algebraic surfaces to blend several given surfaces with GC k continuity is given, and the algorithm is very efficient to find low degree algebraic blending surfaces.In order to control the shape of blending algebraic surface, we write them in the Bernstein Beziér form. Most of B B coefficients can be determined by solving a linear system of equations according to continuous conditions. Some coefficients as free parameters are remained, and can be used to control the shape of algebraic blending surfaces. We require them interpolate and/or least square approximate a collection of some special points. An example is given to illustrate the method and efficiency.At the last section of this paper, a solid modeling of a teapot is presented by eight piecewise algebraic surfaces using our method. The lid is constructed by three piecewise algebraic surfaces of degree four with GC 2 continuity. The body is combined by two piecewise quadratic algebraic surfaces with GC 1 continuity. The spout is made of two piecewise cubic algebraic surfaces with GC 1 continuity, and the handle is a single quartic algebraic surface.