Abstract: This paper presents a new method for constructing approximating subdivision, which is obtained by introducing an offset variable to the interpolating four-point ternary subdivision schemes proposed by Hassan. The new scheme unifies the ternary approximating subdivision and interpolation subdivision. The uniform convergence and C^k smoothness of the limit curve produced by our subdivision scheme are analyzed by means of the Laurent polynomials. It is pointed out that the parameter in the subdivision can be adjusted to control the shape of the limit curve. Numerical experiments show that the proposed method is effective.