Abstract: A method for constructing G² continuous and shape adjustable extension curves and surfaces between two cubic Bézier curves and surfaces is proposed. Firstly, three auxiliary extension curves are designed. Among which the first one meets G¹ continuity at the connection with the two curves to be extended, and the other two meet G² continuity at the connection with one curve to be extended, and G⁰ continuity at the connection with the other curve to be extended. Then, the three auxiliary curves are combined by the potential function to generate the extension curve with shape parameters. Finally, the assignment scheme of shape parameters is determined according to the geometric characteristics of the curve to be extended, and the final extension curve is obtained. The curve construction method is extended to surface, and the connection between two non-adjacent bi-cubic Bézier surfaces is solved. The results of numerical experiments on the curves and surfaces with different geometric features show that the extended curves and surfaces constructed by the proposed method have good smoothness.