The Constructions And Applications Of Subdivision Curves

Subdivision allows to generate smooth curves and surfaces by applying simplerefinement rules to the given control polygon and control mesh, and it has attractedmuch attention and found widespread applications in many areas. This thesis designsseveral effective curve subdivision schemes.Firstly, a ternary four-point approximating stationary subdivision scheme is presentedthat generates theC2limiting curve. The generating polynomial method has been usedto investigate the uniform convergence andC k-continuity of this subdivision scheme.At the same time, using three trigonometric B-spline basis functions, we present aternary four-point approximating non-stationary subdivision scheme. The asymptoticequivalence is used to investigate the convergence and smoothness of the non-stationarysubdivision scheme. Secondly, based on the classical four-point interpolatingsubdivision scheme, a new binary five-point relaxation subdivision scheme is presented.The generating polynomial method is used to investigate the uniform convergence andC kcontinuity of this subdivision scheme. The subdivision scheme generates a familyof C n(n1,2,3,4,5)limiting curves for certain range of tension parameter w. Especially,when w takes some specific values, the limiting curves will turn out to be fractals. Atlast, considering that shape preserving of curve is an important subject in the geometricshape design, we present a five-point binary subdivision scheme with a parameter uwhich is proved to be convexity-preserving. The generating polynomial method and theH lder exponent are used to investigate the uniform convergence andC kcontinuity ofthis subdivision scheme. Moreover, conditions on the initial points are discussed for thelimit curve to be convexity preserving, which are verified by experiments.