Abstract:  In order to obtain the general expression of polynomial potential function which can make the corresponding transition curve reach Ck ( k is an arbitrary natural number) continuity at the endpoints, based on the required conditions of the potential function deduced from the continuity condition, the minimal times of the polynomial potential function is determined according to the number of the required conditions, and the potential function is expressed as a linear combination of the classical Bernstein basis functions which are yet to be determined. According to the function values and derivative values at the endpoints of the Bernstein basis functions, the required conditions of the potential function are converted into an equation set about the undetermined combination coefficients. Solving the equation, we obtain the general expression of the potential function which satisfies the expected continuity condition. Considering that the shape of the transition curve is uniquely determined by the curves needed to be transferred, another new polynomial potential function with one higher degree is constructed. By similar derivation process, the general expression of the new potential function which can make the transition curve reach arbitrary Ck continuity and with shape adjustability is obtained. With the help of the degree elevation formula of the classical Bernstein basis functions, the relationship between the two kinds of potential functions is deduced. The properties of the potential functions and the characteristics of the corresponding transition curves are also analyzed. Legends of the potential functions and the transition curves are put forward and they can verify the correctness of the theoretical analysis results and the validity of the presented method.

Key words: transition curve, potential function, Metaball technology, shape adjustment