Some Research On The Weights Of Rational Bézier Curves

This paper does some work on the weight factors of rational Bézier curves, and gives a method to select the appropriate weight factors. This method also gives the change rule of the weights according to the characteristics of the approximated curve by adjusting parameters. In addition, this paper makes it clear how the changes of the parameters and the weight factors exert the influence to the properties of rational Bézier curves: the shap preserving, local controlling, closing to the control polygons. This paper contains five chapters. Chapter 1 briefly reviews the history of the development and present status of curve and surface modeling in CAGD. Chapter 2 introduces the expression and the properties of rational Bézier curves, and the geometric meaning of quadratic rational Bézier curves. In chapter 3, firstly the exponential-type weight method for constructing rational Bézier curves is introduced with examples, then the circular arc curve is constructed by this method, and the weights of every vertex are computed. Chapter 4 gives a new method to select the weight factors of rational Bézier curves. The rational Bézier curves constructed using this method not only keep all the properties of common rational Bézier curves, but also are more close to the control polygons, and the proof of these properties is given. In addition, this method is simpler and less calculation than the exponential-type weight method. Some numerical examples are given to illustrate the validity of our method, and show its advantage compared with other method.