Interval Implicitization Of Spline Surfaces, Degree Reduction Of Interval Surfaces And Roots Of Interval Polynomials

Conservation of geometric information is very important for Computer Aided Geometric Design(CAGD), but only approximate results can be obtained in many cases for the approximation property of some algorithms and the existence of computer float error. Interval operation is presented to avoid information loss during some geometric procedure. Instead of a point, we use an interval that contains the given point in our computation. By this way, the theoretic accurate result is contained in the computation result and the information loss is avoided.In this paper, our main job is to study interval implicitization of parametric surface, reduction of interval surfaces and the number of "zero" s of an interval polynomial. The importance of error control in CAGD and geometric computation is explained, the history and the state-of-the-art of these problems are reviewed, and then some examples are given to illustrate the significance of interval operation.First, we study the interval implicitization of rational B-spline surfaces. This problem is the generation of the case of curves which is useful to surface operations, such as seeking the intersection of two tangent surfaces. Instead of the method used in the case of curves in which searching the centric curve is followed by searching the boundary curves, we directly seek the boundary surfaces of the interval implicit surface. By introducing the distance, energy and normal direction of the interval surface that determine the geometric shape, we establish an optimization model and give the algorithm of this problem. We give some examples illustrating the algorithm and discuss the application of this method.Secondly, we study degree reduction of interval spline surfaces. Degree reduction overcomes the contradiction between reducing the complexity of geometric process and avoiding the geometric information loss. Degree reduction of tensor product interval spline surfaces, triangular partition interval spline surfaces over polygonal domain and interval PS surfaces are respectively discussed.Thirdly, we study the number of the "zero" s of a univariate interval polynomial. Seeking the roots of polynomials is a very important job, but computer float error restricts its applications. In this thesis, we introduce interval polynomials which avoid the information loss. The definition of "zero" and its multiplicity are presented. Then we generalize Descartes rule, Budan-Fourier theorem and Sturm theorem for polynomials to interval polynomials.Finally, we study the number of "intersection" s of two multivariate interval polynomials and generalize Bezout theorem which determine the number of the intersections of two algebraic curves.