Singularity And Properly Posed Interpolation Problems Of Some Spline Spaces

It is well known that spline is an important approximation tool in computationalgeometry, and it is widely used in many engineering fields. Some special examples ofpolynomial functions appeared in mathematics researches already. For the complexityof object, the research of the multivariate spline has very important significance in thetheoretics and applications. Multivariate splines have connection with univariate splinesin some sense, but they are different in essence, it is not the simple extension of theunivariate splines. Hence, the research results of multivariate splines are not perfect asthose of univariate splines, it still need to be further studied. The thesis is organized asfollows:In Chapter 2, we defineâ–³_MS^μpartition is similar toâ–³_MS partition. For anyμ, the algebraic singularity condition of S_μ+1^μ(â–³_MS^μ) is obtained by the Generator bases inmodules method presented by Professor Z. X. Luo. Wang transformed the structureof spline to corresponding algebraic problem in 1975. In 2001 a mechanical method isproposed to solve the generator bases in modules through the reduced rules by ProfessorZ. X. Luo. It becomes very convenient to study the multivariate spline spaces becausethe generator basis of the conformality equations over an inner vertex are composed ofsome vectors of degree 1 and 0 in general, which are obtained from this method. FromDu's paper, it is easy to find some equivalent relations between the singularity of splinespace overâ–³_MS^μpartition and the intrinsic properties of algebraic curves. Whenμ=2, the geometric singularity condition of S_μ+1^μ(â–³_MS^μ) is proposed. Also, in order to illustrateour conclusions clearly, some examples are given out. All these results are very helpfulfor the further research on the classification and parametrization of algebraic curves andso on.In Chapter 3, using multivariate splines to interpolate scattered points is an important applied field. Bivariate spline spaces have many applications, such as numericalapproximation, surface fitting, scattered points interpolation, multivariate numerical integration, finite element method, numerical solution of partial differential equation, CAGDand computer graphic etc.. Obviously, in order to better understand rnultivariate spline space and to apply it, the first problem is to understand its algebraic structure. We alreadyget perfect results for the case of degree d is much larger corresponding to smoothness r, such as d＞3r+2. However, splines with lower degree are more interested for people inapplications. For example, when r=1, the case of d=2, 3, 4 respectively. But the caseof S₃¹(â–³) is still unknown to us, people can not give its dimension and do not know itsdimension whether depends on the geometric condition of the partition. Hence, it is anopen problem to determine the dimension of S₃¹(â–³) spline space. But S₃¹(â–³) spline spaceis very important to us. Since its simple computation and stability, S₃¹(â–³) is the lowestdegree space in those spaces satisfying the dimensions excess the number of the verticesof triangulations until now. In other words, S₃¹(â–³) space is the lowest degree C¹ spacewhich can interpolate all the vertices in triangulations.For the difficulty of determining the dimension of S₃¹(â–³), people turn to considersome special triangulations. Obviously, it is more useful to find the dimension for general triangulation. In this chapter, we discuss a class of triangulations satisfying someconditions, and study S₃¹(â–³) space on it. First we decompose those triangulations, thenconstruct admissible sets and Lagrange sets recursively, hence their dimensions are determined clearly. Thus, we find these triangulations are nonsingular. In the last, we give amethod to construct triangulations on planar scattered points, such that the generatedtriangulations are exactly in our considered class of triangulations.In Chapter 4, the dimension of bivariate polynomial space P_d is (?) as known.Then what is the dimension of piecewise continuous polynomial space? This question hasimportant significance in studying posed interpolation of spline and so on. Comparingto multivariate polynomial spaces, the dimension of spline space is more difficult. Upto now, many related questions are still open. Triangulation is the common partition inpractical, so its dimension is followed more concern. The study of dimension of bivariatespline space is originated from the guess of dimension of spline space of Strang. Itsrepresentative is L.L. Schumaker who gave the upper bound and the lower bound ofdimension of arbitrary triangulation. It is diffcult to give a general dimension formulafor any triangulation. Because the dimension of spline space not only depends on thetopology property of triangulation, such as the number of vertices, edges and trianglesof triangulation, but also heavily depends on geometric property of the triangulation.In this chapter, we improved the upper bound of dimension for general triangulationby numbering vertices and Smoothing Cofactor method. Moreover, some corollaries areproposed for deciding the dimension of spline space.In Chapter 5, T-mesh is a rectangle mesh with T nodes in essence. T-spline is point based spline defined on T-mesh. Deng etc. presented the definition of T-mesh splinespace, on base of T-spline introduced by Sederberg etc.. T-spline is a tensor productpolynomial in every cell of T-mesh and satisfying smoothness in every interior edge. UsingB-net method, they derived the dimension of T-mesh confined that the smoothness is lessthan half of degree of the spline functions. In this chapter, we use Smoothing Cofactormethod and reordering skill to conformality equations of inner lines, then derive thedimension formula in regular T-mesh spline space. Moreover, the formula also holds forgeneral T-mesh spline space, such as periodic spline space on T-mesh, composite T-meshspline space and T-mesh with holes spline space. Thus our result is more general.