Determination of Tangent Vectors in Construction of Ferguson Interpolation Curves and Surfaces

Abstract

In technical practice we often need to find an interpolation curve which must go through the given base points. A basis for the calculation of the piecewise interpolation curve is the Ferguson cubic curve, the final shape of which is significantly influenced by the magnitude and the direction of the tangent vectors at the startpoints and endpoints of the individual segments. This article describes a method for calculating tangent vectors at every definition point, which ensures a perfect adaptation of the shape of Ferguson cubic curves to the given configuration of the definition points. This method of determining tangent vectors shows minimal undesirable waving among given points, overshooting in the vicinity of given points is considerably limited, and first-degree continuity is ensured among individual parts of the Ferguson cubic curve. The results are used to create a mathematical model of the given surface. The mathematical model is formed by connecting the Ferguson 12 vector patches. A spherical surface was selected as the testing surface, because it is easy to judge the accuracy of the method by comparing the values of the coordinates of the points on the calculated interpolation surface with the exact analytically calculated values.