Progressive Iterative Approximation Property Of Gt-bézier Curves

Computer Aided Geometric Design(CAGD)mainly studies on representation,approximation,analysis and synthesis of curves and surfaces.Curve and surface fitting of scattered data points is an important topic in CAGD,which mainly focuses on interpolation and approximation.Interpolation is to construct curve or surface passing exactly through all given data points.Approximation is to construct curve or surface closing to all given data points under some error metrics.Progressive Iterative Approximation(PIA)is a new method of curve and surface construction of data fitting in recent years,which has the advantages of local interpolation,no need to solve linear equations and obvious geometric significance in iterations.Therefore,it has been attracted much attentions in CAGD.Lin et al in 2005 indicated that the parametric curves or surfaces defined by normalized totally positive bases have the PIA property.Generalized toric-Bézier(GT-Bézier)curve is a kind of parametric curve defined by the generalized toric-Bernstein(GT-Bernstein)basis functions,which are constructed by a given real knot points set and the generalization of classical Bernstein basis functions.In this thesis,we prove that the rational GT-Bernstein bases are normalized totally positive bases,which means that the GT-Bézier curve has the PIA property.And some representative examples are given to verify the PIA property of GT-Bézier curves.Furthermore,we also prove the PIA property of NURBS curves in a new way.The thesis is organized as follows.Chapter 1 introduces the backgrounds of the thesis.The Chapter 2 recalls the basics of Bézier curves,its PIA property,and the proof of the total positivity of Bernstein basis functions.In Chapter 3,the NURBS curves and its PIA property are presented.Additionally,we prove the PIA property of NURBS curves is proved in a new way.Chapter 5 is the main part of this thesis.Firstly,it gives the definitions of GT-Bernstein basis functions,its rational form,and the GT-Bézier curve.Then,we prove the rational GT-Bernstein bases are normalized totally positive bases,which indicates the PIA property of GT-Bézier curves.Finally,the representative examples are given to verify the results.The Chapter 5 concludes the whole thesis.