INTERPOLATION AND APPROXIMATION BY RATIONAL CURVES OF BEZIER AND NURBS-CURVES

Анотація

Relevance. Rational Bezier curves and NURBS curves are widely used in modeling curvilinear objects due to the great flexibility and efficiency of the method. Therefore, it is relevant to develop an interpolation method and approximation by these curves of a discrete series of points both in the plane and in three-dimensional space.

Method. The work is devoted to the development of a new approach to interpolation and  approximations curve fitting, represented by a set of discrete points. The analytical description of the desired curve is implemented using a rational Bezier curve and a NURBS-curve. To solve this problem, two approaches are proposed. The first approach is that the weights of the points are set in advance and then the coordinates of the points of the interpolating or approximating rational Bezier curve as well as the NURBS-curve are calculated. The second approach is that the coordinates of the points are set in advance and then the weights of the control points of the Bezier curve as well as the NURBS-curve are calculated. At the beginning of the process, are set not only coordinates, but also parameters are set to a discrete row of points, that is, each point has the following definition: T (x, y, u) on the plane or T (x, y, z, u) in the three-dimensional space, where u - parameter. To solve the interpolation problem, a system of linear equations is created in which each equation reflects the equality between the analytical formula for a curve and a given point. Moreover, the number of interpolated points cannot be more than the order of the interpolating curve. Thus, we have a system of N linear equations, where N is the number of points equal to the number of points of the curve. Unknown are N control points of the desired curve. Moreover, in the first approach, the unknowns are coordinates of control points, and in the second weights of points.To solve the approximation problem, the Least Squares method is used. In the beginning, a sum of squared functional of the terms of the differences between the analytic formula of the curve and the coordinate of the given point is created. The optimization problem of minimizing this functional is solved. For this, a system of linear equations is created., each equation of which is a derivative of the functional with respect to a given parameter and equated to zero. In the first approach, the desired parameters are the coordinates of points, and in the second weights of points.

Results.Two methods of interpolation and approximation of a point series by rational Bezier curves and NURBS-curves were developed.

 Conclusions. The test cases carried out using computer programs and visualization of results confirm the validity of the proposed methods.