【作者】 张伟红；

【导师】 檀结庆；

【作者基本信息】 合肥工业大学 ， 计算数学， 2006， 硕士

【摘要】 等距曲线又称为位差或平行曲线,它们是基曲线沿法向距离为d的点的轨迹。其在工程中得到广泛的使用,是近十年来CAD/CAM的研究热点之一。由于曲线的单位法矢包含平方根项,等距曲线的代数次数相当高,且一般不再具有原基曲线的相同类型。而且除直线、圆、平面、圆柱面、圆锥面和圆环面以外,有理曲线的等距曲线一般无法表示为有理形式,所以为了使CAD/CAM系统能有效处理,就必须用各种方法对等距曲线进行逼近。目前,等距曲线的逼近方法主要有:(1)等距移动控制网格来得到等距曲线逼近曲线控制网格的方法;(2)基于插值或拟合的方法;(3)基圆包络逼近法。前两种逼近方法几何直观性强,操作相对简单,但缺点是逼近精度比较低,计算存贮量大。第三种方法具有较高的逼近精度,但其等距逼近曲线的次数相对较高,而且通常不再保持原等距曲线的几何性质。 本文从参数速度模的有理逼近着手对平面Bézier曲线的等距曲线逼近进行了研究,给出了基于参数速度模的代数逼近。该方法保证基曲线沿着法矢方向平移定距离,这使得交互操作相当方便,同时利用连分式的递推性能够有效地减少计算量及数据存储量,并具有很好的逼近精度。更多还原

【Abstract】 Offset curves, also called parallel curves, are defined as locus of the points which are at constant distant d along the normal from the generator curves. Offsets are widely used in various engineering applications, which is one of the most important geometric operations in CAD/CAM systems. In general, an offset is functionally more complex than its progenitor because of the square root involved in the expression of the unit normal vector. If the progenitor is a rational curve, then its offset is usually not a rational curve except for special cases. Special cases for curves include straight lines and circles while planes, spheres, circular cylinders, circular cones, toruses and cyclides are the special cases for surfaces. Many methods approximating the offset curves should be used to make CAD/CAM systems operated effectively. At present, the main offset approximation methods are: (1) control polygon-based methods;(2) interpolation methods;(3) circle approximation methods. The first and second methods are simple and look more intuitionistic, but have lower accuracy and larger storage. The third method has a good accuracy, but the disadvantage is a high degree.In this thesis, the algebraic rational approximation algorithm of the planar Bezier offset curve is given, which is based on the approximation of the norm of parametric speed. This method can not only preserve the offset curve moving along the normal direction, which is convenient for the alternate operation, but also decrease the computational complexity and the space of storage by using the recursion property of the continued fraction. At the same time, the method leads to significant improvements in accuracy and efficiency. A numerical example is given to show the effectiveness of the new method.更多还原