On The Theory And Application Of Normalized B-basis

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, affine 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.Normalized B-basis can not only preserve the advantage of traditional surface model, but also represent curves and surfaces in a widely range. So study on normalized B-basis is very meaningful. However, research on normalized B-basis is just beginning and there exist a lot of work to do, such as properties, existence, construction and B-algorithm of normalized B-basis. The theory and application of normalized B-basis are discussed hi this dissertation, major works are as follows:Firstly, sufficient and necessary condition for the existence of normalized B-basison (n+1)-dimensional spaceC'[a,b] is given. Furthermore, the normalized B-basis isdirectly obtained from B-basis.Secondly, based on a basis of the functional space, the sufficient condition for the existence of normalized B-basis is established. An construction method of normalizedB-basis on (n+l)-dimensional space Cn[a,b] is presented. Furthermore, by using this method, the normalized B-basis can be obtained directly from any basis of (n+1)-diniensional spaceCn[a,b].Thirdly, the symmetry of normalized B-basis on space Cn [a, A] is proved.Fourthly, practical construction method of B-algorithm is discussed. For some functional spaces, the B-algorithm of normalized B-basis are given.Lastly, the representation of special curves by normalized B-basis is given, such as cycloid, sinusoidal, catenary and exponential curves. Moreover, using normalized B-basis, C-Bezier surface, H-Bezier surface, H-C-B6zier surface model, and then some transcend surfaces are represented.