三角域上二元三次插值样条函数

Bivariate Cubic Interpolational Spline Function over Triangles

给出一种新的三角域上二元三次插值样条函数,并证明了这种插值函数表达式的唯一性和在整个三角域上是C^1连续的.

The theory and application of interpolational function over triangles have been studied by Farin in 1986[1], Barnhill in 1981[2], and Boehm in 1984[4]. In the author's opinion, Farin's research[1] appears to be the best. A relatively simple method proposed by Farin is the method of cine parameter interpolant. The nine parameter interpolant possesses C^0 continuity. But C^0 continuity does not guarantee the continuity of partial derivatives; thus it can not guarantee smoothness of surface and is in general not applicable in engineering. In order to guarantee C^1 continuity, Farin proposed interpolational function of higher degree and division of triangle into many small triangles. Thus C^1 continuity makes Farin's method very complicated for use in engineering. It is believed that the new method presented in this paper can overcome the shortcomings of Farin's method and still keep C^1 continuity. The key to the success of the new method is the introduction of a new kind of bivariate function-bivariate cubic interpolational spline function. The symmetry of this function makes the form of basic functions simple and reduces the number of basic functions from 9 to 5; thus the author's new method is comparatively simple to use. Farin's method requires many conditions of triangles and subdivisions of triangles to be given, but the author's method requires only a small number of boundary conditions to be given. Thus the author's method is expected to be applicable to design of shape of aircraft and automobiles. Now a few remarks will be made concerning the mathematics of this paper. In this papeer, the basic functions of the bivariate function are determined by the values and partial derivatives of function at vertices of triangles. Thus, the bivariate function can interpolate the function values at the knots of triangluar networks. The author gives proof of C^1 continuity of the bivariate function over triangles and of uniqueness of representation of the function. The author utilizes spline function by which the partial derivatives of function at vertices of triangles are obtained.