Bézier作图定理与三次Bézier曲线的几何特征

P.E.Bézier利用速端曲线,苏步青和刘鼎元运用仿射变换,都对Bézier曲线的几何特征作了深入研究。Bézier在文献1]中提出用几何作图求Bézier曲线上的点及其切线的方法即Bézier作图定理。本文从作图定理入手详细地分析了平面三次Bézier曲线的几何特征,指出图2所示的λ、μ（或）是决定几何特征的一对不变量,给出了λ、μ（或）全平面图（见图3）,进而讨论了空间三次Bézier曲线的某些几何特征。

BEZIER'S PLOTTING THEOREM AND GEOMETRICAL CHARACTERISTICS OF CUBIC BEZIER CURVES

In this paper, taking the plotting theorem as the point of departure, we analyze in detail the geometrical characteristics of plane cubic Bezier curves, including whether a cusp （ a cusp of class one） or one inflexion point or two inflexion points exist on the （ 0, 1 ）; whether double point occurs on 0, 1 ) or ( 0 , 1 ] and whether the curve is convex or not.The geometrical characteristics of plane cubic Bezier curve can be determined uniquely by two parameters λ , μ or λ,μ（see Fig. 2）on the diagram（fig. 3）. The single inflexion curve in Fig. 3 represents the cases when the curve can be transformed into general cubic polynomial. The single inflexion region indicates the cases when the curve has only one inflexion point on （ 0, 1 ）and another is not on （ 0, 1 ）.We may obtain the parameter uc of cusp, uI of inflexion point and u1, u2 of double point.By using plotting theorem we can also make the conclusion that a space cubic Bezier curve has not cusp, double point and its spiral direction doesn't change.