In order to enhance the shape representation flexibility and simplify the smooth connecting conditions of Bézier curve and surface, three groups of polynomial basis functions with shape parameter were constructed. Based on them, three kinds of new curves and surfaces, which had the similar structure with the quadratic, cubic and quartic Bézier curve and surface respectively, were defined. The new curves and surfaces not only posess the basic properties of Bézier curve and surface, but also have shape adjustability. Furthermore, under simple conditions, the composite curves and surfaces consist of the new curves and surfaces achieve G2 or G3 continuity. Besides, the method of constructing curves tangent to the given polygons is presented. This method is simple and effective. And the curves are conformal to the given polygons.