【作者】 刘刚；

【导师】 王国瑾；

【作者基本信息】 浙江大学 ， 应用数学， 2012， 博士

【摘要】 多项式标准全正(NTP)基混合控制顶点构成的NTP参数曲线曲面是计算机辅助几何设计和几何造型的基本工具之一.本文着重研究了NTP曲线曲面的逼近和插值问题.1.NTP曲线的约束逼近.对常见的NTP曲线,Said-Bezier型广义Ball曲线和Delgado-Pena曲线,给出了一种统一算法实现低阶的曲线近似表达高阶曲线.利用NTP多项式基函数和单变量Jacobi多项式之间的转换关系以及Jacobi多项式的正交性,把逼近问题转换为最小二乘问题,从而计算出逼近曲线的控制顶点.降阶算法有L2范数下误差最小,端点高阶插值,一次性降多阶,降阶曲线显式表达,误差先验估计等优点.逼近方法简单快捷,因此将在CAD系统中的数据通讯、数据压缩等方面有重要的应用价值.2.NTP曲线曲面的渐进迭代逼近.在算法的收敛范围内,对常见的NTP曲线,张量积曲面和三角参数曲面,给出了带权渐进迭代逼近的显式精确解.对于两种NTP基,即Said-Bezier型广义Ball基或者Delgado-Pena基,给出相应的基于、(?)andermonde矩阵显式逆矩阵的插值曲线曲面的矩阵解.算法避免了矩阵求逆,所以在逆向工程中有重要的应用价值.3.精确计算NTP-Vandermonde矩阵并用于数据点插值与拟合.给定区间(0,1)内l(l≥n)个单调递增的节点,n次Said-Bezier型广义Ball基函数在这组节点下的配置矩阵：Said-Bezier-Vandermonde矩阵是严格全正矩阵.对这一类NTP-Vandermonde矩阵,给出了双对角分解的公式化结果,及计算双对角分解矩阵的快速算法.算法具有高度的精确性,且降低了诸多运算的复杂度,比如用于平面点列插值,相应的线性方程组的求解复杂度可从O(n3)降到O(n2).通过一些应用实例,如求线性方程组的解,求矩阵特征值以及最小二乘拟合数据点,验证了算法的正确性和精确性.4.弦长参数化.给出了一种新的参数化方法,使得参数化后的Bézier曲线的参数尽可能地接近弦长参数.这个问题的解最终归结为求一个一元二次方程的根.对于一般的Bezier曲线和有理二次Bezier曲线,给出了弦长参数化的精确的显式解,而对于高次的有理Bezier曲线则利用复合辛普生积分公式给出了数值解.更多还原

【Abstract】 Normalized totally positive (NTP) parametric curves and surfaces constructed by polynomial NTP basis blending control points are one of basic tools for computer aid-ed geometric design and geometric modeling. This manuscript focuses the main attention on the approximation and interpolation problem of NTP curves and surfaces.1. Constraint approximation of NTP curves. For common NTP curves, Said-Bezier type generalized Ball curves and Delgado-Pena curve, we present an unified algorithm to approximate these curves by lower degree curves. By using the transformation relation be-tween NTP polynomial basis function and univariate Jacobi polynomial and the orthonor-mality of Jacobi polynomials, the approximation problem can be simplified to a least square problem, hence the control points of lower curve can be computed. This algorithm has sev-eral advantages, such as approximation error in the L2norm is minimal; high continuities at the endpoints of the two curves; multi-degree reduction once; explicit approximation expressions; error bound computed in advance and so on. The algorithm is simple and fast and hence will gain extensive application in data communication, data compression of CAD systems.2. Progressive iteration approximation (PIA) for NTP curves and surfaces. In the range of weighted PIA algorithm convergence, we give out the explicit exact PIA solutions for common NTP parametric polynomial curve and surface, and triangle Bezier surface. For the two NTP basis, Said-Bezier type generalized Ball basis and Delgado-Pena basis, the matrix solution of corresponding interpolation curves and surfaces are given based on explicit inverse matrix of Vandermonde matrix. This results avoid the tedious calculation of the inverse matrix and hence will gain extensive application in reverse engineering.3. Accurate computations with NTP-Vandermonde matrices and the application in interpolating and fitting given data points. Given l(l≥n) increasing nodes in the interval (0,1), the collocation matrix for degree n Said-Bezier type generalized Ball basis at these nodes:Said-Bezier-Vandermonde matrix is a strictly totally positive (STP) matrix. For this type NTP-Vandermonde matrix, we present the bidiagonal decomposition formula, and fast algorithm to compute the bidiagonal decomposition matrix. The algorithm has high accuracy and low complexity for many operations. For example, the bidiagonal decomposi-tion can be used to compute the linear system generated by Said-Bezier basis interpolating given data. The complexity can be reduces from (?)(n3) to (?)(n2). By some application such as solving Said-Bezier linear system, computing eigenvalues of Said-Bezier-Vandermonde matrix, and least square fitting data points, we demonstrate the correctness and high accu-racy of the algorithm.4. Chord-length reparametrization. We present a new reparametrization method to make the parameter of parameterized curve as close as possible to chord-length param-eter. The solution of this problem can be analytically reduced to finding the unique real root on (0,1) of a quadratic equation. The exact and explicit solutions of chord-length reparametrization are presented for Bezier curves and rational quadratic Bezier curves, whereas the numerical solutions are given by Composite-Simpson’s integral formula for higher degree rational Bezier curves.更多还原