A class of cubic parameter curves are given. Hermite curve, Ball curve, Bezier curve and Timmer curve are its particular cases. Under certain conditions, this cubic parameter curves have the following properties: Convexity, retaining convex form, contact with the second edge of characteristic polygon and so on. This curve is different from Bezier curve, because its, shape can be changed after its characteristic polygon is given, so its shape can be changed according to our actual requirements.