【作者】 陈军；

【导师】 王国瑾；

【作者基本信息】 浙江大学 ， 应用数学， 2010， 博士

【摘要】 曲线曲面是计算机辅助几何设计(CAGD)系统中的基本工具,CAGD的大多数操作都是以曲线曲面为对象的.而无论是根据给定的几何信息构造满足几何约束条件的曲线曲面,还是为压缩几何信息的数据量而近似合并曲线曲面,它们都是在实际生产中被广泛应用的操作,因而一直成为人们关注的热点之一.本文围绕这两类问题展开了深入的研究,取得了以下丰富的创新性成果：1.四阶均匀α-三角/双曲多项式B样条曲线的保形插值：基于几何约束中位矢约束的曲线造型,其实质上就是构造插值所有给定点的曲线.而保形插值,就是使得插值曲线能够保持住型值点的外形特点.构造四阶均匀α-三角/双曲多项式B样条曲线的核心思想是,把一个参数化的奇异多边形与三角/双曲多项式B样条按某一个形状因子调配,自动生成带形状参数且插值给定平面点列的C2或G1连续的三角/双曲多项式B样条曲线.它既继承了均匀三角/双曲多项式B样条曲线的特点,也继承了奇异混合样条插值曲线在不要求解方程组或进行繁复的迭代的前提下进行插值的优点.为使每条与形状参数相应的插值曲线都能保单调或保凸,只需把曲线一阶导矢的两个分量或者曲率符号函数分别转化为类Bernstein多项式,从而利用二次Bernstein多项式的非负性条件,简单快捷地得到形状参数α保证曲线保单调或保凸的取值范围.2.规避障碍物的G2连续低阶样条曲线的构造：以基于几何约束中位矢约束的曲线造型对应的形状因子为临界值,得到能够规避障碍物的形状因子的范围.首先,对由线段构成的,能够规避障碍物的引导多边形进行光顺,得到G2连续的样条曲线.既给出了这种样条曲线的有理二次参数形式,又给出了隐函数形式.其主要思想是首先对引导多边形进行改进,插入部分中点以作为新的控制顶点.然后根据位矢约束求解每一段曲线的形状因子,并对所有的形状因子进行比较,取最大的一个来构造整条曲线,使之能够规避所有障碍物的凸包,并保持G2连续.与以往方法相比,本文构造的曲线具有以下优点：1.次数较低,却仍能够保证曲线整体G2连续；2.保形性良好,曲线与引导多边形具有相同的拐点；3.无需解高次方程,直接计算就可得到结果；4.控制多边形直观可见,便于对曲线进行控制.特别地,三次泛函样条曲线还可进行局部调整,但仍能保持G2连续.最后列举了多个数值实例,用来验证算法的简单与有效.3.三角Bezier曲面修改与调整方法：提出了一种基于几何约束中位矢约束和法向约束的三角Bezier曲面修改与调整方法.调整后的曲面满足多个参数点处位矢和相应法矢向量的几何约束.在角点无约束或者角点处边界曲线高阶连续的约束条件下,通过Lagrange乘子法,分别得到不同的调整曲面,使得距离函数在L2范数下达到最小.该算法简单有效,适用于各类CAD系统的交互设计.4.曲线的近似合并：讨论了两类曲线,B样条曲线的近似合并以及有理Bezier曲线的区间近似合并.对于B样条曲线,利用极值条件,通过求解一个线性方程组,使得距离函数在L2范数下达到极小,合并曲线的控制顶点可用矩阵显式表达,同时原曲线与合并曲线间距离函数的L2范数也可以精确得到.然后这个方法被成功地推广到两相邻非均匀B样条曲面的近似合并以及多段非均匀B样条曲线的一次性近似合并上.最后,利用齐次空间和二次规划问题,还探讨了非均匀有理B样条曲线的近似合并,同样得到了很好的结果.对于有理Bezier曲线,首先利用顶点摄动法,使得摄动误差在某个范数下达到最小,得到两条有理Bezier曲线的多项式近似合并曲线,以此作为区间曲线的中心表达形式.然后利用已有的计算结果直接得到区间长度固定的误差曲线,或者利用二次规划得到逼近效果更佳的区间长度不固定的误差曲线,两种方法都可以通过中点离散技术进行优化.如果对误差进行限制,还可以得到端点插值的合并区间曲线.5.三角Bezier曲面的近似合并：基于三角Jacobi基的正交性,以及其与三角Bezier基之间的基转换矩阵,得到两张或四张相邻m阶三角Bezier曲面与所求n(n≥m)阶近似合并三角Bezier曲面的距离函数的L：范数.然后分别在角点无约束或者角点处边界曲线高阶连续的约束条件下,通过最小二乘法分别得到不同的合并三角Bezier曲面,使得距离函数在L2范数下达到最小.合并曲面的控制顶点可用矩阵显式表达,同时原曲面与合并曲面间距离的L2范数也可以精确得到.特别地,通过提高合并三角Bezier曲面的次数可减小合并误差,改善合并效果.该方法计算简单直接,适用性强,逼近效果佳.更多还原

【Abstract】 Curves and surfaces are common tools in CAGD systems, and the most operations in CAGD are based on Curves and surfaces. Both the shape modeling with geometric constraints which are from the given geometric information, and the approximate merging of curves or surfaces have become current research hotspots. In this dissertation, we have made deeply researches on the two topics and provided abundant and innovative results as follows:1. Curve modeling with points constraints means to get a curve to interpolate the given points. And the shape-preserving property is very important for the interpolation curves. In order to get the cubic uniformα-trigonometric/hyperbolic B spline curve, we blend a parametrized singular polyline and the trigonometric/hyperbolic polynomial B-spline curve using a blending factor, to automatically generate a C2 or G1 continuous hyperbolic polynomial B-spline with a shape parameter, which interpolates the given planar data points. By converting the first derivative or the curvature sign function of the interpolating curve into Bernstein polynomial, the nonnegativity conditions of Bernstein polynomial can be used to get the range of the shape parameter a and the necessary and sufficient conditions for the monotonicity or the convexity-preserving property of interpolation curves. The method is simple and convenient, need not to solve a system of equations or recur to a complicated iterative process.2. An algorithm for finding a G2 continuous, obstacle-avoiding curve in the plane is presented. Based on the guiding polyline path, both the rational quadratic parametric spline curve and the implicit functional splines curve are obtained. First, we partition the guiding polyline into control polygon sections by inserting several midpoints of polyline. Then, we find respective shape parameter of each curve section to avoid the vertices of the convex hull of an obstacle. At last, we choose the biggest shape parameter to avoid all the obstacles. Comparing with previous methods, the curves constructed by our approach have the following advantages:1. it is G2 continuous but with low degree; 2. it is shape-preserving, and the number of inflection point is the same as the one of the guiding polyline path; 3.it is obtained directly, and we need not to solve the fourth order equations; 4. the control polygon is visual, and we can adjust the curve easily. Specially, all shape parameters of cubic functional splines curve are local, so that they can be adjusted respectively with G2 continuity. Finally, several examples demonstrate the effectiveness and validity of the algorithm.3. A new algorithm for shape modification of triangular Bezier surface is presented by point constraints and normal vector constraints. Without any constraints or with the boundary continuity constraints at three corners, the new surface satisfies specified geometric constraints, such as multiple points and normal vector of the selected parametric point on the given triangular Bezier surface.With the help of Lagrange multipliers, the L2 norm between the surfaces before and after modification is minimized. The numerical examples show that this method is convenient for interactive design in CAD systems.4. Two kinds of approximate merging algorithms are presented:the approximate merging of two adjacent B-spline curves by one B-spline curve and the approximate merging of a pair of rational Bezier curves by interval Bezier curve. Applying the distance function between two B-spline curves with respect to L2 norm as the approximate error, we investigate the problem of approximate merging two adjacent B-spline curves into one B-spline curve. This method can be extended to the approximate merging problem of multiple B-spline curves and of two adjacent surfaces very easily and successfully. After minimizing the approximate error between curves or surfaces, the approximate merging problem can be transformed into equations solving. So we can obtain both the new control points and the precise error of approximation explicitly in matrix form. On the other hand, based on the center curve and error curve of the interval Bezier curve, the approximate merging of a pair of rational Bezier curves by interval Bezier curve was obtained. The basic idea is to get the polynomial Bezier curve as the center curve by using the perturbation theory first. Then, we compute the error curve with constant or unconstant interval by solving linear equations or solving a quadratic programming problem. Both of the two error curves can be improved by the application of the well-known subdivision approach to this method. Furthermore, the interval Bezier curve can interpolate the rational curves at the two end points with the boundary constraints.5. Based on the orthonormality of triangular Jacobi polynomials and the transformation relationship between triangular Jacobi and Bernstein polynomials, the distance function between the original degree m triangular Bezier surface and the degree n(n≥m) approximate merging of 2 or 4 neighbouring triangular Bezier surface with respect to L2 norm was obtained. Without any constraints or with the boundary continuity constraints at three corners, we got the optimal approximate merging triangular Bezier surfaces by the least square method respectively, simultaneously the distance function reached the minimum. Both the control points of approximate merging surface and the precise error of approximation were expressed explicitly in matrix form. The algorithm is simple and direct, applicable for most cases. The degree elevation could reduce the merging error. Several numerical examples are presented to illustrate the correctness and validity of the algorithm.更多还原