带两个形状参数有理二次三角Bézier曲线

Rational Quadratic Triangular Bézier Curves with Two Shape Parameters

本文提出了带两个形状参数的有理二次三角Bézier曲线,由4个控制顶点生成的曲线具有传统有理三次Bézier曲线的几何特性,包括端点性质、对称性、凸包性、几何和仿射不变性、变差缩减性.分析了在权因子固定情形下,通过改变形状参数值可以局部调控曲线形状;也得出当形状参数值都为-1时,曲线可退化为直线段.曲线在适当的控制顶点下,可精确表示椭圆弧和圆弧,从而可方便整圆的表示.在控制顶点和权因子相同的条件下,当形状参数取值在一定范围内,曲线具有比有理三次Bézier曲线对控制多边形更好的逼近.

A rational quadratic trigonometric Brzier curve with two shape parameters was presented in this paper, The curves generated by four control points had the geometric characteristics of the traditional rational cubic Bezier curve, including the properties of the endpoint interpolation, symmetry, convex hull property, geometric ＆ affine invariance, and variation diminishing property. It was analyzed that when the weight factors was fixed, the shape of the curve was controlled locally by changing the shape parameter value, also was concluded that when the shape parameter values were -1, the curve was degenerated into a straight line segment. The elliptical arcs and arcs at the appropriate control points were accurately represented by the curve, so that the whole circles was expressed conveniently. Under the same control vertex and the weight factor, the curve had a better approximation to the control polygon than the rational cubic Brzier curve when the shape parameter was taken in a certain range.