一类三角Bézier曲线的形状特征

Shape Features of a Kind of Trigonometric Bézier Curve

在几何造型的许多应用中,良好的曲线形状应该消除不必要的奇点和拐点,因此往往需要预知与分析参数曲线的各种形状特征,以避免出现奇异形状的设计风险。为了快速确定参数曲线的形状特征,利用锥面的齐次性简化了参数曲线的形状条件,得出了一类带2个形状参数的二次三角Bézier曲线的尖点条件锥和2张重结点边界条件锥;3张特征锥面及其切平面将特征空间划分为不同的特征区域。曲线的形状特征完全由特征点在特征空间的分布区域决定。用垂直于坐标轴的平面切割特征空间,可得到基于包络与拓扑映射方法的所有形状条件分布图。进而讨论了形状参数变化对各特征区域的影响,相关结果可使设计者明确如何配置控制顶点或者调节形状参数,使得生成曲线为全局凸或局部凸曲线,或具有所需要的奇点与拐点,或将当前曲线形状调节为另一种所需的形状。

In many applications of geometric modeling,curves of desirable shape should eliminate the unnecessary singularities and inflection points.Therefore,to avoid potential risk in shape design,it is essential to predict and analyze the shape features of parametric curves.In order to quickly determine the shape features of parametric curves,the shape conditions of the parametric curve are simplified due to the homogeneous property of cones,and the cusp conditional cone and two boundary loop conditional cones are obtained for a quadratic trigonometric polynomial Bézier curve characterized with two shape parameters.These three characteristic cones and their tangent planes divide the characteristic space into different characteristic regions.The curve’s shape features are completely determined by the distribution region which the characteristic point locates in the characteristic space.It is shown that the shape diagrams obtained by the method based on the theory of envelopes and topological mappings can be derived from characteristic space by virtue of planar slices,which are vertical to one of the axes.Furthermore,the influences of shape parameters on the associated characteristic regions are also discussed.The obtained results enable the user to place control points or choose shape parameters so that the resulting curve is globally or locally convex,possessing wanted singularities or inflection points,or enjoying the desired shape features.