Summary: A new four-point ternary interpolating curve subdivision scheme and a cubic B-spline curve subdivision scheme with parameters are proposed. By using the two proposed methods, a ternary curve subdivision scheme which blends interpolating and approximating is obtained. This blending subdivision scheme can unify the interpolating subdivision and the approximating subdivision. A geometric interpretation of this blending subdivision scheme is given, and its continuity is analyzed. We generalize this method to the surface case. We propose a 1-9 interpolating surface subdivision scheme and a tensor product cubic B-spline surface subdivision scheme on the quadrilateral mesh, and use them to obtain a kind of ternary surface subdivision scheme that blends interpolation and approximation. The continuity of this scheme is analyzed. The numerical example shows that the method is reasonable and effective.