平面NURBS曲线及其Offset的双圆弧逼近

The Biarc Approximation of Planar NURBS Curve and Its Offset

除直线、圆弧、速端曲线等少数几种曲线外 ,平面参数曲线的 offset曲线通常不能表示成有理参数形式 ,因此在实际应用中 ,为了方便造型系统中数据结构和几何算法的统一表示 ,offset曲线通常用低次曲线逼近来表示 .通过用双圆弧逼近表示 NURBS( non- uniform rational B- spline)曲线及其 offset,并利用双圆弧逼近的特有性质 ,把 offset的双圆弧逼近转化为原曲线的双圆弧逼近 ,简化了问题的求解 .同时考虑了双圆弧逼近算法中分割点的选取、公切点的确定以及误差估计等主要问题 .具体算法在自主开发的 Gems5.0中实现 .经实例表明 ,算法稳定。

The planar offset curve cannot be expressed as rational parametric curve in gene ral excepta few types of curves such as line, arc, Hodograghs etc. In practice, the offset curve usually is approximated by lower degree rational polynomial cur ve in order to have the unified expression of data structure and geometric algor ithm in the commercial modeling systems. In this paper, an approximation approac h to NURBS (non-uniform rational B-spline) curve and its offset is presented b y using biarc. The biarc approximation of offset curve is simplified to biarc ap proximation to original curve. Some important and key problems, such as the reas onable selection of split points in NURBS curve, the definition of cotangent poi nt of biarc and error estimate method, are discussed. Examples verify the effici ency and reliability of the algorithms, which are implemented in the commercial geometric modeling systems Gems5.0 developed by CAD Center of Tsinghua Universit y.