APPROXIMATION BY RATIONAL SURFASES OF BEZIER AND NURBS-SURFASES

Abstract

Relevance. Rational Bezier surfases and NURBS-surfases are widely used in modeling curviliniar objects due to the great flexibility and efficiency of the method. Therefore, it is sense to develop an approximation method by these surfases

Method. The work is devoted to the development of a new approach to approximation surfases, represented by a set of discret points. The analytical description of the desired surfases is implemented a rational Bezier surfases and a NURBS-surfases. To solve this problem, two approaches are propozed. The first approach is that the weights of the control points are set in advance and then the coordinates of the points of the approximation rational Bezier surfase as well as the NURBS-surfase are calculated. The second approach is that the coordinates of the control points are set in advance and then the weights of the control points of Bezier surfase as well as the NURBS-surfase are calculated. At the beginning of the process, are set only coordinates, but also parameters are set to a discret points, that is, each points has the following definition: T(x,y,z,u,v) in the three-dimensional space, where u,v – parameters. To solve the approximation problem, the least squares method is used. In the beginning, a sum of squared functional of the term of the differences between the analytic formula of the surface 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 derivatev of the functional with respect to a given parameter and equated to zero. In the first approach, the desired parameters are coordinates of points, and the second weights of given.

Results. Tho methods of approximation of a point series by rational Bezier surfase were developed.

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

Keywords: approximation, uftional Bezier surfaces, NURBS- surfaces.