摘要
Bézier曲线和曲面是CAD/CAM系统中广泛使用的造型工具。参数曲线和曲面的降阶逼近已经成为CAGD领域的一个热点研究问题,它主要研究用低次的形式来近似逼近给定次数的曲线或曲面。降阶逼近在几何设计上有很多的应用,如数据交换、数据压缩和数据比较等。本文围绕着CAGD中Bézier曲线和三角Bézier曲面的降阶逼近问题进行了深入的研究,其主要成果及创新点如下。
首先,传统的降阶方法都只考虑了曲线的参数连续,而我们利用曲线的几何信息来研究逼近问题,并首次引入了G~2连续的约束。因此,曲线在端点的位置、切向和曲率大小在降阶前后能够保持一致。然后,我们提出了一个全新的方法来解决Bézier曲线在L_2范数下的最佳逼近问题,它可以通过最小化曲线的L_2误差这一目标函数而得到。为了避免近似曲线在端点附近出现奇异性问题,我们在目标函数里加入了修正项。此外,对于G~1连续约束这一特殊情形,我们提出了另外一种基于二次规划方法的新算法,并用线性约束来满足切向在端点的一致性。此时的逼近问题就转变成一个带线性约束关于两个参数的二次规划问题。我们可以应用新算法使近似曲线的参数化更加接近于弧长参数化。
其次,区间[0,1]上的多项式曲线可以表示成不同基的形式,如Bernstein和第二类Chebyshev多项式的形式。我们给出了Bernstein基和第二类Chebyshev基之间的变换矩阵,通过它们能实现Bernstein系数和Chebyshev系数的转换。接着,我们分析了基变换的稳定性,结论是:Chebyshev-Bernstein基变换矩阵的l_1和l_∞条件数随着次数n的增长速度远小于power-Bernstein基变换的情形,并且速度很接近(稍快于)Legendre-Bernstein基变换的情形,所以也是良态的。利用基变换矩阵,我们提出Bézier曲线在加权(t-t~2)~(1/2)平方范数下最佳的降多阶逼近方法。并将它推广到保端点r阶和s阶连续(r,s≥0)的情形,尽管不是最佳的,但是提供了一个很好的近似。该方法具有O(n~2)的计算复杂度。我们建议对次数过高的曲线不要使用基于基变换的降阶算法,因为此时这类方法很可能是不稳定的。另外,我们还估计了逼近的L_1误差的上界,它是后验的。
最后,对于给定一张n次三角Bézier曲面,我们研究了带边界约束的用更低次数为m的三角Bézier曲面来近似逼近它。提出对曲面的三个角点进行约束,使得边界曲线在每个端点能保持C~α连续。利用约束最小二乘法最小化曲面的l_2和L_2距离,我们提出两种逼近算法来得到降阶后曲面控制网格的矩阵表示。这两种方法都能应用于连续拼接的三角曲面片以及与曲面细分结合使用时的情形,结果生成的近似曲面片是整体C~0连续的。我们还给出了逼近的误差估计,并举例说明方法的有效性。
Bézier curves and surfaces are widely used modelling tools in CAD/CAM systems.Degree reduction of parametric curves and surfaces,which tries to approximate a given curve or surface of certain degree by another one of lower degree,has become an important and hot problem in CAGD.It has many applications in geometric modelling,such as data exchange,data compression and data comparison.In this thesis,we have made a systemic theoretic research on the problem of degree reduction of Bézier curves and triangular Bézier surfaces in CAGD.The main creative results are as follows.
Firstly,in contrast to traditional methods,which typically consider the components of the curve separately,we use geometric information on the curve to generate the degree reduction.And the constraint of G~2-continuity is introduced for the approximation problem,so positions,tangents and curvatures are preserved at the two endpoints.We then present a novel approach to consider the multi-degree reduction of Bézier curves in L_2-norm.The optimal approximation is obtained by minimizing the objective function based on the L_2-error between the two curves.In order to avoid the singularities at the endpoints,regularization terms are added to the objective function.Furthermore,for the special case of G~1-continuity,we presents another approach to solve the approximation problem in terms of the quadratic programming method,with linear constraints to satisfy the coincidence of tangents at the endpoints.Then,degree reduction is changed to solve a quadratic problem of two parameters with linear constraints.We apply the new approach to improve the parameterizations of approximating curves to be close to arc-length parameterizations.
Secondly,a polynomial curve on[0,1]can be expressed in terms of Bernstein polynomials and Chebyshev polynomials of the second kind.We derive the transformation matrices that map the Bernstein and Chebyshev coefficients into each other,and examine the stability of this linear map.In the p=1 and∞norms, the condition number of the Chebyshev-Bernstein transformation matrix grows at a significantly slower rate with n than in the power-Bernstein case,and the rate is very close(somewhat faster) to the Legendre-Bernstein case.Using the transformation matrices,we present a method for the best multi-degree reduction with respect to the(t-t~2)~(1、2)-weighted square norm for the unconstrained case, which is further developed to provide a good approximation to the best multidegree reduction with constraints of endpoints continuity of orders r,s(r,s≥0). This method has a quadratic complexity,and may be ill-conditioned when it is applied to the curves of high degree.We estimate the posterior L_1-error bounds for degree reduction.
Finally,for a given triangular Bézier surface of degree n,we investigate the problem of approximating it by a triangular Bézier surface of degree m with boundary constraints.We constrain continuity conditions at the three corners of triangular Bézier surfaces,so that the boundary curves preserve endpoints continuity of any orderα.The l_2 and L_2 distances combined with the constrained least-squares method are used to get the matrix representations for the control points of the degree reduced surfaces.Both methods can be applied to piecewise continuous triangular patches or to only a triangular patch with the combination of surface subdivision,the resulting piecewise approximating patches are globally C~0 continuous.Also,we estimate the error of approximation and provide some examples to demonstrate the effectiveness of the two methods.
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