An Extension Of Rational Bezier Curve And Its Application

The representation method of curves and surfaces are an important part of the study of Computer Aided Geometric Design, and the choice of basis functions has important influence on the nature of the curves and surfaces. The Bernstein polynomials are basis function of the Bezier curves and surfaces, which has been widely used in CAGD fields. But for given control points, the Bezier curve is uniquely determined. In order to enable to modify the Bezier curve’s shape locally, Bezier curves are extended by introducing the shape parameters and trigonometric basis functions, which have achieved some good results. In CAGD, rational Bezier curves are the most basic modeling tools, but the shape of a rational Bezier curve cannot be modified locally too, because moving the location of control points or changing the value of weights, will cause the shape change of the whole curve. Therefore, combined with shape parameters which have the ability of the local modification on the curve shape, this paper proposed a new curve construction method — the rational cubic trigonometric Bezier curve with two shape parameters, RCT-Bezier curve, for short. The related properties and applications of the RCT-Bezier curves were discussed.This paper consists of the following six parts. The first part introduced the background and the development history of the curve and surface modeling, and the current research status of the extended curves. The second part reviewed the definition and the properties of Bezier curves and rational Bezier curves, and the continuous stitching conditions between the curves. The third part mainly introduced a class of polynomial functions with multiple shape parameters, and the construction of cubic extended Bezier curve based on the functions (CE-Bezier curve for short), and discussed the smooth connection conditions of the CE-Bezier curve and uniform quadratic B-spline curve. In the fourth part, we proposed the new rational cubic trigonometric Bezier curve with two shape parameters, whose geometric properties are similar to the traditional rational Bezier curve. The presence of shape parameters provided more flexible control on the curve’s shape than that of traditional Bezier curve. Moreover, the weights offered an additional control on the curve’s shape. Using the given end point curvatures, we derived the conditions which the shape parameters and weights must satisfy such that the generated curve always lies in the convex hull of its control polygon. Then we demonstrated RCT-Bezier curve can accurately represent elliptic arc and circular arc by numerical experiments, and discussed the relationship between RCT-Bezier curve and the traditional rational Bezier curve with the same control points and weights. The fifth part presented the C2 and G2 continuity conditions of splicing two pieces of RCT-Bezier curves, and designed the vase, flower petals and other special patterns with the piecewise RCT-Bezier curves. The sixth part summarized the full text and put forward the future work.