参数曲线造型中保形理论与算法的研究
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摘要
保形插值是几何设计的一项基本技术,在自由型曲线曲面造型、数值逼近以及逆向工程等领域中都有重要的应用价值.然而现有的保形插值方法都还存在或多或少的不足,如:绝大多数方法都只适用于函数型点列,真正适用于参数型点列的方法很少;或都需要通过求解方程组或繁琐的迭代过程或求解一个最值问题,才能得到保形的插值曲线.有鉴于此,本文在深入研究了曲线曲面造型中的各种保形理论与算法的基础上,提出了几类新的保形插值的方法,解决了已有保形插值方法存在的不足,主要成果及创新点为:
1.利用奇异混合的思想构造出一类带有形状参数的多项式样条曲线,即均匀α-B样条曲线,在无需反求方程组、迭代过程和最值求解的情况下就能轻松做到插值给定的点列;它含有形状参数α,这就增加了曲线的柔性,同时又和原B样条曲线具有相同的参数连续性.再利用Bernstein多项式的正性条件,找到形状参数α的取值范围,使得当且仅当参数α在该范围内时,相应的整条α-B样条插值曲线都是保单调的,且为C~2连续;其次,为实际使用方便并提高算法效能,对α-B样条插值曲线的每段曲线,选取不同的参数,使整条曲线也是保单调的,并达到G~1连续.
2.进一步研究了上述均匀α-B样条曲线的保凸插值问题.首先推导出该类曲线的相对曲率的表达式;然后对每一个整体凸的点列,利用相对曲率不变号的准则找出形状参数α的统一的或分段的取值范围,使得与该范围内每个形状参数相应的插值样条曲线都是保凸的.接着,将上述结论推广到分段凸点列的情形,最终得出一个方便快捷的自动保凸插值方法.
3.为了能通过调整节点间距来改变连续阶,使得曲线上有尖点或夹有直线段,给设计千姿百态的复杂曲线提供可能.我们接着引进了非均匀α-B样条曲线,并相应地研究其保形(保单调和保凸)插值的可能性与算法.对单调点列,得到了非均匀α-B样条插值曲线保单调的充要条件;对凸点列,在一定的条件下得到曲线无拐点和尖点的充分条件.
4.为了弥补普通三角多项式样条曲线在形状调整方面的不足,同时又考虑到多项式样条曲线不能精确表示一些超越曲线,我们利用奇异混合的思想构造了一种新的带有形状控制参数的三角多项式样条曲线.在无需反求方程组、迭代计算和最值求解的前提下,即可产生插值给定点列的C~2或G~1连续的一族三角多项式样条曲线;进一步研究了该类曲线的保凸性,得到插值曲线保凸时形状参数的取值范围.该方法较好地解决了超越曲线的保形插值问题.
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.
1.利用奇异混合的思想构造出一类带有形状参数的多项式样条曲线,即均匀α-B样条曲线,在无需反求方程组、迭代过程和最值求解的情况下就能轻松做到插值给定的点列;它含有形状参数α,这就增加了曲线的柔性,同时又和原B样条曲线具有相同的参数连续性.再利用Bernstein多项式的正性条件,找到形状参数α的取值范围,使得当且仅当参数α在该范围内时,相应的整条α-B样条插值曲线都是保单调的,且为C~2连续;其次,为实际使用方便并提高算法效能,对α-B样条插值曲线的每段曲线,选取不同的参数,使整条曲线也是保单调的,并达到G~1连续.
2.进一步研究了上述均匀α-B样条曲线的保凸插值问题.首先推导出该类曲线的相对曲率的表达式;然后对每一个整体凸的点列,利用相对曲率不变号的准则找出形状参数α的统一的或分段的取值范围,使得与该范围内每个形状参数相应的插值样条曲线都是保凸的.接着,将上述结论推广到分段凸点列的情形,最终得出一个方便快捷的自动保凸插值方法.
3.为了能通过调整节点间距来改变连续阶,使得曲线上有尖点或夹有直线段,给设计千姿百态的复杂曲线提供可能.我们接着引进了非均匀α-B样条曲线,并相应地研究其保形(保单调和保凸)插值的可能性与算法.对单调点列,得到了非均匀α-B样条插值曲线保单调的充要条件;对凸点列,在一定的条件下得到曲线无拐点和尖点的充分条件.
4.为了弥补普通三角多项式样条曲线在形状调整方面的不足,同时又考虑到多项式样条曲线不能精确表示一些超越曲线,我们利用奇异混合的思想构造了一种新的带有形状控制参数的三角多项式样条曲线.在无需反求方程组、迭代计算和最值求解的前提下,即可产生插值给定点列的C~2或G~1连续的一族三角多项式样条曲线;进一步研究了该类曲线的保凸性,得到插值曲线保凸时形状参数的取值范围.该方法较好地解决了超越曲线的保形插值问题.
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.
引文
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