双三次曲面的反演算问题

Inversive Calculation of Bicubic Curved Surface

在科技领域中广泛应用的双三次曲面,一般都按给定曲面特征点(又称控制点)求解,然而在不少实际问题中,例如地层层面的模拟,都是反过来,即知道曲面上的数据点,求解该曲面的特征点,以确定该曲面方程。这是曲面的反演算问题,本文讨论双三次曲面反演算问题,推导出反演算的数学表达式,依特定值找出系数矩阵,并用二次曲线拟合法求出曲面片角点的切矢量,依方程解出曲面的特征点,从而可确定通过离散数据点的曲面方程。

Bicubic curved surface is widely used in scientific and technological field. The curved surface is usually defined according to its characteristic points. But in quite a few practical situations, the characteristic points are not or not quite on the bicuzbic curved surface. The simulation of laminated surface is an example of such case. It is a problem of inversive calculation, i. e. the characteristic points of the curved surface are calculated on the basis of the data points so as to establish the equation Of the curved surface. This paper discusses the inversive calculation of bicubic curved surface. It derives the mathematical expression of the inversive calculation. The coefficient matrix is found on the basis of the given conditions. The tangential vector at patch corners have been solved by fitting of quadric curve. The characteristic points of the curved surface are calculated by the expression and the expression of the curved surface passing through the discrete points can thus be established.