Necessary and Sufficient Conditions for the C² Continuous Limit Curves and Surfaces of 4-Point Interpolatory Subdivision Scheme

The continuity of 4-point interpolatory subdivision scheme was studied. One of Dyn's theories was proved in a new way by using Jordan's normal form. Necessary and sufficient conditions for the second derivative of limit function were obtained, and expression of second derivative was acquired as well. The result was accordingly extended into the case of surfaces.