Research On Shape Preserving Theory And Algorithm In Parametric Curve Modeling

Shape preserving interpolation is an essential technique in geometric design and of great significance in various areas such as curve and surface modeling,numerical approximation and reverse engineering. However, the existing methods on shape preserving interpolation also have some drawbacks such as: most of them can only generate some function-form shape preserving interpolating curves which are unaccommodated with the parametric curves, commonly used in CAGD systems, or some of them must solve a system of equations or a minimum problem or recur to a complicated iterative process to obtain a shape preserving interpolating curve. After detailedly investigating various shape preserving theory and algorithm in curve and surface modeling, this thesis aimed to propose several new shape preserving interpolating methods, and avoid the shortcomings in existing shape-preserving interpolating methods. The main creative results of this thesis are as follows:1. A new kind of parametric polynomial spline with a shape parameterαis constructed by linear singular blending technique. We call them uniformα-B spline. Then we can conveniently make them interpolate the given data points but dispensing with solving a system of equations or a minimum problem or going at any iterative process. These curves contain a shape parameter which can adjust the softness of curve and have the same continuity with the original B-spline. In succession, using the positive conditions of Bernstein polynomial to find a range in which the shape parameter takes its value so as to make the corresponding interpolating curves monotonicity-preserving and C~2 continuous. Then in order to make the algorithm more convenient for application and improve its efficiency, we can let the each segment of uniformα-B spline take respective shape parameter and the whole uniformα-B spline interpolating curve is also monotonicity-preserving and G~1continuous.2. The convexity-preserving interpolation problem of uniformα-B spline is also explored. First, the relative curvature expression of uniformα-B spline interpolating curve is deduced. Then letting the sign of relative curvature keep unchanged for a global convex data points, we find a range in which the shape parameterαtakes its value so as to make the corresponding interpolating curves convexity-preserving. The similar algorithm can also be gained for the cases of piecewise convex data points.3. For the sake of changing the order of continuity of the interpolating curve so as to produce cusps and straight-line segments on it, we introduce non-uniformα-B spline. Then the shape (monotonicity and convexity)-preserving algorithm of this curve is also explored. For monotone data points, we obtain the necessary and sufficient condition for the non-uniformα-B spline interpolating curve monotonicity-preserving; for convex data points, we obtain the sufficient condition for the nonuniformα-B spline interpolating curve being out of inflection points and cusps under some hypothesis.4. In order to make up the deficiency of ordinary trigonometric polynomial curves in aspect of shape adjustment, then in view of polynomial spline cannot represent some transcendental curves, a new kind of parametric trigonometric polynomial spline with a shape parameter is constructed by linear singular blending technique. Interpolating trigonometric polynomial parametric curves with C~2 (or G~1) continuity can be automatically generated without having to solve any system of equations or do any iterative computation. Then, the convexity of the constructed curves can be guaranteed by the appropriate value of the shape parameter. This method settles the shape-preserving interpolation problem of transcendental curves more successfully.