Abstract: A new method for extending cubic Bézier curve is presented in this paper. G² continuity is used to describe the smoothness of two cubic curves at their joint point, which offers two additional freedom degrees for adjusting the shape of the extended curve, thus the unadjustability of cubic curves in C² continuity is removed. The shortest arc-length, minimum energy and minimum curvature variation of the curve, are used to set up the objective functions, respectively. The freedom degrees of the extended curve are determined by minimizing the objective functions. The comparison of the curves by different objective functions is included.