Curve Subdivision Algorithm And Its Properties

Because the curve subdivision method can produce well-performed curves, the research on curve subdivision method has become a hot topic currently, and has attracted much attention in both academic community and industry. In view of this, this paper presents three kinds of effective subdivision algorithms, and studies the smoothness, the convexity and other properties of the subdivision algorithms.Firstly, we present a binary approximating subdivision scheme based on interpolation subdivision scheme by integrating the classical interpolation subdivision scheme with approximation subdivision scheme. And the convergence and continuity are discussed by means of generating polynomials. Secondly, we present a five-point binary relaxation subdivision scheme with single parameter, which is an asymmetric interpolating subdivision scheme constructed by using the method of generating polynomials, the uniform convergence and continuity of the scheme are analyzed, and the convexity preserving property is proved. Although in the form of asymmetry, this scheme is of high smoothness, and can get very good effect in some asymmetric modelling. Since shape-preserving has long been an important research topic in geometric modeling, we finally study the convexity preserving property of the five-point binary subdivision scheme. Given strictly convex initial control points, we discuss how to choose the parameter such that the limiting curve can preserve convexity. Our theoretical analysis has been verified by numerical examples.