Multivariate Rational Splines On Triangulations And Their Applications

Multivariate rational spline is the natural generalization of multivariate polynomial spline. However, the space of rational spline is more complicated, so the theories are not as perfect as those of polynomial case. Some questions should be further studied. This dissertation studies multivariate rational splines denned on the triangulation. We focus on C~1 rational spline defined on the planar triangulation. Moreover, we consider the application of it in CAGD. The thesis is organized as follows:Chapter 2 studies the range restricted interpolation problem. First we discuss the equivalent expression of C~1 rational spline in barycentric form. Non-Uniform Rational B-spline(NURBS) is powerful and have great potential in the definition of shape. In 1994, for the exchange of industrial product data, International Standard Organization(ISO) issued STEP standard, which makes NURBS a unique mathematical method for the definition of industrial product. In the paper [1-3], author construct the rational spline defined on the planar triangulation by generalized wedge function method and give the equivalent hybrid expression, which is local, explicit and good at shape preserving. And as what we have said above, NURBS method is perfect and widely used. So it's significative to study the connection between rational spline and NURBS. Based on the above idea, we discuss the equivalent expression of C~1 rational spline in barycentric form and build the bridge between rational spline and NURBS, which is the combination of three cubic Bernstein-Bezier triangular patch. And Bernstein-Bezier triangular patch has been studied since Farin proposed systemically in 1980. It has developed rapidly and been widely used, which will accelerate the development of rational spline when we apply them into the research of rational spline.Further based on the the expression of C~1 rational spline in barycentric form, the range restricted scheme is presented. In CAGD, a general problem is to construct the interpolant with a certain degree of continuity. When there are some properties inherent in the data, like positivity, monotonicity and convexity, one wishes to preserve. In CAD, the user may think it's good that the interpolant preserves some properties. And in practice, some physical properties can be described in mathematical form. When the data arise from a physical experiment, like densities and rainfall, where negative values are not physically meaningful. It is vital that the interpolant preserves nonnegativity. Such problem is called nonnegative-preserving interpolation. The more general problem is range restricted interpolation, i.e.. the interpolant is constrained by the surfaces as same as the data. This problem has been widely studied. The constrained surface has been developed from plane to cubic polynomial surface as the lower or upper or both the lower and upper bound. We give the restricted conditions here on the coefficients of C1-rational spline to ensure the nonnegativity of it. By generalizing this method, we get the range restricted interpolant with lower and upper constraint polynomial surfaces up to degree three. The method is completely explicit. There are no continuous equations and minimization problem included. With the scaling factor, the method we get is a local method. It is flexible and convenient to compute.Chapter 3 discusses the problem of scattered data interpolation on the sphere by using generalized wedge function method. The problem for constructing functions on the sphere arises in many applications, such as geodesy, geophysics and meteorology etc.. where the basic models all belong to this problem. Generalized wedge function method is general for the problem of rational spline functions. So except for 2D space, it is easy to solve the problem in 3D or higher dimension space. Based on this, for some special low dimension manifolds, we get the generalized wedge functions corresponding to them by restricting the ones defined on the high dimension manifolds on them. It's a local method that avoids the difficulties in constructing the interpolants on the low dimension manifolds directly. First for the circular case, the scattered data on the circle and the center of it constitute a disc. By restricting the generalized wedge functions defined on the triangulation of the disc on the circular, we get the ones corresponding to the vertices of each arc. Based on the same idea, we get the C1 interpolant defined on the sphere. Scattered data on the sphere and the center of the sphere constitute a ball. By restricting the generalized wedge functions defined on the tetrahedron on the spherical triangle, we get the ones corresponding to the knots of each triangle, then the interpolant is constructed.Chapter 4 gives a data-dependent method of nonsingular self-adaptive triangulation for ^(A) and 63(A) multivariate spline spaces. Given a set of points in R2, it is well known that the triangulation of this set of data is necessary in many scientific computation field, such as surface design and fitting, definite element and some other large scientific computation. For the mass data got from practical problem, there is no need of all the data points taking part in the triangulation if we consider from the point of view of efficiency and error. So it's important to study the effective algorithm of optimizing and data-dependent triangulation. It is well known that multivariate splines are important and effective tools for the above scientific computation. But the problem of multivariate spline space has not been completely solved. Particularly for the lower degree multivariate spline space, the dimension of it depends not only on the topological property of partition, but also on the geometric property of partition. Lower degree multivariate splines are widely applied, for many of it'sproperties have been known and it's computed conveniently. So it's significative to study the triangulation with the following particular properties:1. add data points into the triangulation as few as possible and improve the accuracy of rational spline surface defined on this triangulation effectively;2. Sl(A) and ï¿¡3(A) are non-singularity over the constructed triangulation;3. satisfy the traditional local optimal condition.Based on the above consideration, and the rational spline defined on the planar triangulation have some good properties, such as local and explicit. Based on the algorithm proposed in paper [5], which gives a non-singularity triangulation method for ï¿¡2 (A) and S3 (A) spline spaces, we present here a triangulation method which satisfies the above three properties by using the equivalent form of C1 rational spline presented in chapter 2. By using the coefficients of (^-rational spline, we define a discrete norm for it. It is a discrete method. Then the definition of weight for each knot is given, which is a measure of the importance of the knot in the representation of the rational spline. The numerical examples show the method feasible and effective.