Total Positivity Of Generalized Ball Bases & Conversion Between Triangular And Rectangular SBGB Patches

The thesis is composed of four chapters.The first two chapters are focused on the Said-Ball and Wang-Ball curves, including their definitions, properties and explicit conversion formulas to the Bezier representation, the author also points out that the Said-Ball basis is NTP. The emphasis of the next two chapters are laid upon the two new families of the generalized Ball bases which are introduced by Wu HongYi, i.e., Said-Bezier type generalized Ball (SBGB)bases and Wang-Said type generalized Ball (WSGB)bases, the curves constructed on these bases include the Bezier curve, the Said-Ball curve, the Wall-Ball curve and some intermediate curves. These curves have many the properties similar to the ones of the Bezier curve.By making a further study of SBGB curves, the author acquires two main results, namely:1. Using a corner-cutting algorithm, this thesis shows that the SBGB bases are normalized totally positive (TNP), and the WSGB bases are not TNP.2. This thesis also presents an explicit formula that converts a triangular SBGB patch of degree n to a degenerate rectangular SBGB patch of degree nxn by reparametrization and the SBGB dual bases.