线性p-Bézier曲线的几何形状

Geometric Shape of Linear p-Bézier Curve

为了丰富非多项式空间的拟Bézier系统的几何性质,针对线性三角多项式空间的p-Bézier曲线进行研究.首先通过几何变换与参数变换将椭圆的参数方程化为p-Bézier形式;然后通过对比,指出除去退化情况外线性p-Bézier曲线必为椭圆弧;再给出该椭圆的中心,焦点,长、短轴顶点这些几何元素与其控制顶点间的关系式;最后给出了线性p-Bézier曲线为圆弧的充要条件.实例结果表明,文中的几何元素可以通过控制顶点的线性插值得到.

To develop the geometric properties of the Bézier-like systems for non-polynomial spaces,researches are done for p-Bézier curves of the linear trigonometric polynomial space.Firstly,a parameter equation of an ellipse is changed into the p-Bézier form from geometric transformations and parameter transformations.Secondly,according to comparison,a conclusion that every linear p-Bézier curve except degenerated cases is an elliptic arc is drawn.Thirdly,the relations between the geometric elements of the ellipse,such as the center,foci,vertices of major and minor axes,and the control points are given.Lastly,a sufficient and necessary condition under which a linear p-Bézier curve is an arc is given.Moreover,example analysis shows that these geometric elements can be represented as linear interpolation forms of control points.