Some Researches On Approximate Implicitization And Piecewise Algebraic Varieties

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.