Abstract: 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.