Research On The Existence Of Rational Interpolants And Normalized B-basis In CAGD

Rational function interpolation theory and its application are an important part in research on rational approximation. There have been a lot of achievements in uniqueness, algorithms, error estimate and etc., especially in algorithms. But there doesn't always exist rational interpolation function for arbitrary interpolation conditions given in advance. Moreover, other results such as uniqueness, algorithms and error estimate are given which bases on that rational interpolation function exists. If the existence can't be settled well, the determinacy of these results will be influenced. Normalized B-basis, namely optimal normalized totally positive basis, plays an important role in CAGD, for it possess positive properties such as variation diminishing, convex-hull, acne invariance, tangency to the control polygon at the endpoints and B-algorithm. The widely used basis functions in CAGD, such as Bernstein, B-spline and NURBS basis, are all normalized B-basis.In this thesis, we discuss the existence of the rational interpolation and normalized B-basis in CAGD. This thesis consists of five chapters. In chapter 1, we not only retrospect the background of the research on rational interpolants and normalized B-basis, but also retrospect the study actuality of the existence of rational interpolants and normalized B-basis .In chapter 2, through analyzing the unattainable point of Thiele fractions interpolation , the method for testing the unattainable points is given. Then we give two metheds of changing the unattainable points into attainable points.The chapter 3, we give two kinds of algorithms of dividing interpolation nodes into subsets for Thiele-Werner interpolation. An efficient algorithm for computing bivariate lacunary rational interpolation is constructed.The chapter 4, we mainly discuss the properties, existence, construction and B-algorithm of normalized B-basis.The chapter 5, we give the normalized B-basis in a kind of rational space, and discuss the properties and applications in CAGD.