Research On The Shape Preserving Of Spline Curves

With the development of CAD technology, there are more and more methods to construct curves and surfaces. Correspondingly, the quality demand is much higher and higher.For example, the curve obtained by fitting some data points not only keeps the smooth but also needs to preserve some mathematical properties including the monotone and convexity and so on. But the usual interpolation method either does not meet the above requirements or is too complex to compute. The improved quasi-interpolations, however, can solve this problem well. Besides, we often come across the problem of shape preserving for a given tangent polygon. For this issue, the T-B spline can give better solutions.This thesis mainly investigates the constructures of quasi-interpolant functions on the basis of uniform B-spline and quartic T-B spline with all edges tangent to the given tangent polygon.It does some study as follows:Firstly, for the given data points {(xi,f(xi))}ni=1on the function f(x), the class of shape preserving quasi-interpolant function (ωAf)(x) is constructed with the cubic uniform B-spline,its control points are the linear combination of three adjacent discrete values of f(x).We can directly get the approximation curve (ωAf)(x) without calculating large-scale equations and prove the conditions under which (ωAf)(x) is linear-rebuilding,monotone and convexity preserving.In addition, we get the multivariate quasi-interpolant function based on the Univariate case. Some numerical examples illustrate that we can get a better approximate curve (ωAf)(x) by choosing parameters appropriately or adding some data points.Secondly, based on the quintic uniform B-spline, we present the quasi-interpolant function(ωBf)(x) whose control points are the linear combination of five adjacent discrete values of f(x).The conditions under which (ωBf)(x) is quadratic-reproducing,monotone and convexity preserving are obtained Similarly, we extend (ωBf)(x) to the two-variate situation naturely.It proves that the approximation effect of (ωBf)(x) is better than (ωAf)(x).Thirdly, through analysis of the properties of quartic T-B spline curves, a planar piecewise quartic T-B spline curve with all edges tangent to the given control polygon is proposed. All control points of the curve segments can be computed by the vertices of the given tangent polygon.The quartic T-B curves are C3continuous and shape-preserving for the polygon and the local modification can be completed by adjusting some parameters. Finally, some numerical experiments illustrate that thealgorithm given in this dissertation is effective.