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This paper uses the continued fraction technique to construct a nonstationary 4-point ternary interpolatory subdivision scheme, which provides the user with a tension parameter that effectively handles cusps compared with a stationary 4-point ternary interpolatory subdivision scheme. Then, the continuous nonstationary 4-point ternary scheme is analyzed, and the limit curve is at least C 2 -continuous. Furthermore, the monotonicity preservation and convexity preservation are proved.

In this paper we present a non-stationary binary three-point approximating subdivision scheme which can generate a wide variety of C3-continuous limit curves with different initial control parameter α0. The continuity of the scheme is proved by using the conclusion in Dyn and Levin, (1995) 5. The advantage of the scheme is illustrated by a number of examples.

This paper presents a new curve algorithm called four point interpolatory-corner cutting subdivision for generating curves that interpolate some given vertices and approximate the other vertices. By four-point interpolatory-corner cutting subdivision, only the vertices specified to be interpolated are fixed and the other vertices are updated at each refinement step. Compared to the four-point interpolation subdivision scheme, it is expected to have improved behavior, and compared to the cutting smooth subdivision, it is able to generate curve interpolating some given vertices. It is more suitable for practical application. The refinement rules are derived to ensure that the eigenvalues of the refinement matrix satisfy the necessary condition of C1continuity. Four-point interpolatory-corner cutting smooth subdivision also contains tension parameters, it is conducive to regulate the shape of limit curve.