【作者】 吴金明；

【导师】 王仁宏；

【作者基本信息】 大连理工大学 ， 计算数学， 2007， 博士

【摘要】 开展隐式曲线曲面的研究在计算机辅助几何设计和几何造型中既有深刻的理论价值又有广泛的应用前景。本文主要针对参数曲线曲面的近似隐式化和分片代数簇的某些问题展开研究。主要工作如下：第二章我们讨论了参数曲线曲面的近似隐式化。参数曲线曲面和隐式曲线曲面是计算机辅助几何设计和几何造型中两种常见的表示形式。将参数形式的曲线曲面转化为代数形式的曲线曲面的过程称为隐式化。然而，精确隐式化(尤其是曲面的精确隐式化)在几何造型中没有得到广泛的应用。这主要是参数曲线曲面的精确隐式化过程很复杂而且不一定可以实现，再加上隐式曲线曲面的阶数很高并且具有不希望的奇异点和多余分支，从而会引起计算的不稳定性和几何造型中拓扑结构的不一致性，这就大大限制了精确隐式化在实际中的运用。为了解决上述问题，我们提出了如下三种近似隐式化算法：首先，我们利用二阶二次代数样条曲线对参数曲线进行近似隐式化，所得到的代数样条曲线不会产生多余分支和自交点，并且具有良好的误差估计和逼近性质。其次，我们利用Multiquadric拟插值和径向基神经网络，提出了参数曲线近似隐式化的另一种方法。该方法具有保形性好，光滑度高，逼近性能好，样本节点数据少等优点。最后，考虑到很难将上述两种方法直接推广到参数曲面的近似隐式化，我们利用紧支集径向基函数作多元散乱数据插值的技巧提出了参数曲面近似隐式化的一种算法。第三章我们讨论了分片代数簇的某些问题。分片代数簇作为多元样条的公共零点集合，是经典代数簇的推广，它不仅和许多实际问题如多元样条插值，CAD和CAGD等有关，而且还为研究经典代数几何提供理论依据。因此，研究分片代数簇是很重要的。首先，讨论了分片代数簇的维数性质，通过引入分片代数簇完全相交的概念讨论了分片代数簇的维数与其定义方程组个数的关系．其次，简要讨论了分片代数曲线的奇点性质。再次，为了有效计算分片代数簇，我们讨论了凸多面体内任意维代数簇的计算问题。通过添加超平面技巧将Groebner基方法应用到凸多面体内任意维代数簇的计算上，从而把凸多面体内的代数簇转化为另外一组多项式方程组的正解，并且得到了该代数簇在凸多面体内的极小分解．最后，基于B-样条系数的Descartes符号准则和Bézier曲线的de Casteljau算法，我们给出了一元样条实根分离算法，也就是计算出一列不相交区间，使得每个区间恰好只包含此样条函数的一个实根。第四章我们主要构造了一类具有紧支集的无穷次可微径向函数。众所周知，Gauss分布函数是一类广泛应用于多元插值和径向基网络的正定径向函数。它具有非常好的逼近效果和指数衰减性质。对Gauss分布函数离中心远处截断，可以马上得到紧支集径向基函数。然而，这样得到的紧支集径向基函数用于多元插值和函数逼近显然是不连续的。因此，结合Gauss函数的特点对其进行改进，我们构造了一类具有可控自由参数的紧支集无限次可微函数。在对自由参数一定的约束条件下，此类函数能够有效地应用到多元函数逼近和多元散乱数据插值中。更多还原

【Abstract】 Implicit surfaces play an important role in many theoretic fields and applied fieldsin Computer Aided Geometry Design and Geometry Modeling. In this thesis, we mainlystudy some problems on approximate implicitization and piecewise algebraic varieties.Our primary work is organized as follows:In chapter 2, we discuss the problem of approximate implicitization of parametriccurves/surfaces. It is well known that parametric curves/surfaces and implicit curves/-surfaces are two important topics in Computer Aided Geometry Design and GeometricModeling. The procedure of transform the parametric form into algebraic form is calledimplicitization. However, accurate implicitization (especially surface implicitization) hasnot been popular in practice. This is due to the fact that exact implicitization of para-metric curves/surfaces always involves complicated computation and its exact implicitform usually cannot be computed. Moreover, the degree of implicit curves and surfacesis higher and implicit curves and surfaces may have singular points and unexpectedcomponents, which lead to computational instability and topological inconsistency ingeometric modeling.In order to solve this problem, we propose the following three algorithms to deal withapproximate implicitization. Firstly, we use quadratic algebraic spline with smoothnesstwo to tackle approximate implicitization of parametric curves. The resulting approxi-mate curves not only don’t have unwanted components and self-intersections, but alsohave good error estimate and approximation behavior. Secondly, we propose a new ap-proach to solve approximate implicitization of parametric curves based on radial basisfunction networks and multiquadric quasi-interpolation. This approach possesses theadvantages of shape preserving, better smoothness, good approximation behavior andrelatively less data etc. Lastly, we propose a method to solve approximate implicitizationof parametric surfaces based on multivariate interpolation by using compactly supportedradial basis functions. In chapter 3, we discuss several problems on piecewise algebraic varieties. As thezeros of multivariate splines, the piecewise algebraic variety is a generalization of theclassical algebraic variety. It is important to study the interpolation by multivariatesplines and algebraic geometry etc. Firstly, we discuss the relationship between thedimension of piecewise algebraic varieties and the number of their defining equations, aswell as several dimension properties by introducing the concept of complete intersectionof piecewise algebraic varieties. Secondly, several singular point properties of piecewisealgebraic curves are discussed. Thirdly, the intersection problem of piecewise curves andsurfaces boils down to the computation of piecewise algebraic varieties. In order to solvepiecewise algebraic varieties, we propose a new method to compute an algebraic varietyon a convex polyhedron by adding hyperplanes with the method of Groebner bases. Thus,the algebraic variety on the convex polyhedron is transformed to the positive solutionsof a system of polynomials. Besides, the minimal decomposition is also obtained. Lastly,we present an algorithm to isolate real roots of a univariate spline, i.e., computing asequence of disjoint intervals such that each of them contains exactly one real root of agiven spline function, which is primarily based on the use of Descartes’ rule of signs withits B-spline coefficients and de Casteljau algorithm of B(?)zier curve.In chapter 4, a kind of multivariate compactly supported infinitely differentiablefunctions is constructed. It is well known that the Gauss distribution function is awidespread used positive definite radial basis function in multivariate scattered datainterpolation. Moreover, it possesses good global approximation behavior and decaysexponentially. It is clear that we can generate the radial functions with compact supportby cutting off Gauss function at large distances from the centers. However the resultingradial function interpolation (approximation) is obviously discontinuous. Thus, we con-struct a kind of compactly supported radial functions which have infinite differentiableproperty for any space dimension with two free parameters. Under certain conditions onparameters, they can be applied to function approximation and scattered data interpo-lation.更多还原