A method of constructing a C ~ surface which interpolates to a function and derivative function on an arbitrary triangle is developed. The calculations involved are simple and efficient, and its polynomial precision set includes all polynomials of degree four or less. The approximate errors of the interpolation function are estimated. An example is presented.
surface modelling, C j surface, triangulation
The problem of constructing a surface based on three dimensional position date (xi, Yi, zi) is currently of active interest to various groups. The applications of this problem are not only in computer geometry-aided design, but also in such fields as mining, geology, cartography, finite element analysis and medicine I . One method of solving this problem is to triangulate the given set of points (xi, Yi) on the xy plane, followed by interpolation on the subtriangles.
The purpose of this paper is to present a method for such a representation of a curved surface with an arbitrary triangle.
Many useful rational i~terpolation methods have been presented by Barnhill 2'3 and Gregory 4. In these methods the interpolation function is evaluated using several interpolation operators along parallels to the sides of the triangle. Usually, transformations are needed to map arbitrary triangles onto the standard triangle s . The method described in this paper is simpler, and may apply to arbitrary triangles without transformation. The error estimation and the example show that the interpolant produces faithful replicas of the original functions.