CUBIC B-SPLINES: PRACTICAL APPLICATION

Abstract

Two constructive algorithms for forming cubic B-splines passing through given points with tangents indicated at these points are proposed. The first algorithm provides a composite cubic B-spline curve passing through n reference points. The composite curve is formed by 3(n-1) elementary cubic segments. At the reference points, the curvature of all segments is zero. Controlling the shape of the curve (with given tangents at the reference points) is achieved by moving auxiliary control points along the given tangents. The second algorithm allows the formation of a cubic B-spline that passes only through two predefined points, but with the ability to control the shape of the spline by moving arbitrarily specified control points. The cubic B-spline equations are derived using two, three, and four control points. For the case of “two control points,” a wellknown result is obtained: the equation of the B-spline identically coincides with the equation of the cubic Bezier curve. For the case of “three control points,” the B-spline equation consisting of two segments is obtained. For the case of “four control points,” we obtain the equation of a cubic curve consisting of three segments. The proposed algorithms enable users to practically construct cubic B-spline curves using the equations derived in the article. The resulting equations allow, without using the Coxde Boer recurrent formula, the calculation and construction of cubic B-splines with three or four control points, respectively. The article presents the basic theoretical information about cubic B-spline curves necessary to solve these problems. Practical examples of constructing composite cubic B-spline curves are provided. The tasks considered in the article can be solved in any graphics editor using NURBS curves, which have become a standard tool for computer modeling. However, this graphical tool does not provide a mathematical model of the curve in the form of equations. On the contrary, the algorithms considered in the article ensure the derivation of the equations of the constructed curves.