2m次贝塞尔曲线降一次逼近及误差分析

2m Degree Bzier Plane Parametric Curves 1 Order reduction Approximation

在计算机图形学、计算机辅助几何设计、计算机辅助制造和计算机辅助设计领域中 ,贝塞尔曲线降次逼近是一个基本而重要的课题 ,它在减少系统数据存储量、增加系统稳定性和提高计算效率等方面有着重要应用。通过对 2 m次平面参数贝塞尔曲线降一次逼近问题的分析研究 ,给出了用 2 m- 1次贝塞尔曲线逼近 2 m次贝塞尔曲线的“封闭”的计算公式 ,推广了文献 [1]中给出的降一次逼近时的误差估计公式 ,并得到了“封闭”的形式。为 CAD系统的用户和计算机图形学、计算机辅助几何设计、计算机辅助制造和计算机辅助设计领域的研究人员使用计算逼近曲线控制顶点和逼近误差的封闭形式提供了方便。而且对于事先给定的容许误差 ,利用文中的方法 ,借助于贝塞尔曲线离散分割算法可以很容易求出满足要求的逼近曲线。

Order reduction of Bezier curves is an important and basic problem in Computer Graphics,CAGD or CAD/CAM systems.In this paper,1-order reduction of even order plane parametric Bezier curves is investigated to reduce the data storage in Computer Graphics,CAGD or CAD/CAM systems,and to increase the robustness and efficiency of Bezier curves calculation.In this paper,the author generalizes the formulae of generating an approximation of order2m-1to a(2m)th order Bezier curves,and gives a formula of computing error in1-order reduction of a plane parametric Bezier curve and the original curve from the result of Chen Jiuping.The representations in the closed form for the coefficients and the error bound are very useful to the users of CAD systems.By using the error bound in the closed form,a simple subdivision scheme for C 1 -constrained and end-constrained order reduction of a plane parametric curve is put forward,and the numerical result is compared visually to that of the best order reduction method.Then,the first subdivision points of the inflection are discussed.Finally,in the method presented in this paper,the order reduction of odd Bezier curves can be easily generalized.