Researches On The Theoretical And Applications Of Multivairate Splines

It is well known that function is the basic object of study in mathematics, and continuous function is an important kind of function. By Weierstrass approximation theorem, any continu-ous function which defined on closed interval can be approximated by polynomials. However, polynomial meets with troubles in practical application due to its strong globality of the polyno-mial. Therefore, piecewise polynomial, i.e. spline function emerges at the right moment. In1946, I.,J. Schoenberg systematically studied and established the basic theory of univariate splines. Re-searches on splines became more and more widely since then. With the deepening of study on spline functions, univariate splines are no longer appropriate to solve many complex practical problems, it is necessary to investigate the multivariate spline function. In1975, Renhong Wang established a new approach to study the basic theory on multivariate spline function on arbitrary partitions using the methods of function theory and algebraic geometry, and presented the so-called the smoothing cofactor-conformality method. Fruitful results of multivariate splines have been obtained up to now. The spline functions have been a fundamental facility of mathemat-ical review and engineering applications. In this paper, we discuss the related theory of spline functions and their applications. It is mainly about the instability of spline spaces over T-meshes with T-cycles, the B splines of S41(△lmn(1)) on3-D1-type tetrahedron partition, the signed distance function approximation of plannar closed curves, the implicit curve fitting with spline functions and the surface reconstruction of scattered data.The main results of this dissertation can be summarized as follows:1. In Chapter1, some background information about multivariate spline functions, curves and surfaces modeling are introduced.2. In Chapter2, we study the instability in the dimensions of spline spaces over T-meshes by using the smoothing cofactor-conformality method. The modified dimension formulas of spline spaces over T-meshes with T-cycles are also presented. Moreover, some examples are given to illustrate the instability in the dimension of the spline spaces over some special T-meshes.3. In Chapter3, the spline space on3-D tetrahedron partition in four-directional mesh is dis-cussed. We obtained the B splines of S41(△lmn(1)) on3-D1-type tetrahedron partition by using the smoothing cofactor-conformality method. The properties of the B splines are also stud-ied.4. In Chapter4, the signed distance function can effectively support many geometry process-ing tasks such as smoothing and shape reconstruction since it provides efficient access to distance estimates. We propose an adaptive method to approximate the signed distance func-tion of a smooth curve by using polynomial splines over type-2triangulation. The trimmed offsets of the curves are also studied.5. In Chapter5, we consider the problem of scattered data curve fitting in Euclidean space, we use bivariate B spline to reconstruct curves implicitly. For closed curves, we use the method of reconstructing curves by spline functions by reconstructing the signed distance function of curves. For general curves, we use the least squares fitting of piecewise algebraic curves with the normal vectors, tangent vectors, and energy constriction of curves.6. In Chapter6, we study the problem of multivariate scattered data surface fitting in three dimensional Euclidean space, we use multi-level spline quasi-interpolation to reconstruct surfaces.