Injecitivity Identification Of Rational Bézier And NURBS Curves/Surfaces

Rational Bézier curves/surfaces and NURBS(Non-Uniform Rational B-Splines)curves/surfaces play an important role in Computational Geometry,Computer Aided Geometric Design(CAGD),Computer Aided Design(CAD)and some related fields.The injectivity of curves/surfaces,which implies curves/surfaces have no self-intersections and the computations of self-intersections have important research significance in offset design,solid modeling,image mor-phing,computer animate,iso-geometric analysis,shape modification and 3D deformation.And injectivity identification is necessary for computation of self-intersections of curves and surfaces.The toric surface is a kind of multisided parametric surface which is derived from algebraic ge-ometry and combinatorics.The toric surface preserves some properties of Bézier surfaces,since its basis functions are the generalization of classical Bernstein basis functions.Thus,the toric surface becomes a popular tool in curve/surface design and the research on its geometric prop-erties becomes a research hot spot.In this thesis,based on the toric degeneration theory of toric surface,we present the definitions of well-posed control polygons/control points sets,and give the sufficient and necessary conditions of injectivity of rational Bézier curves/surfaces and NURBS curves/surfaces.We also present algorithms for the well-posedness identification of the control polygon/control points set.In Chapter 1,we recall the history and the backgrounds of parametric curves and surfaces briefly,including rational Bézier curves/surfaces,NURBS curves/surfaces and the toric surface.We illustrate the injectivity conditions of curves/surfaces,together with their applications in geometric design and modeling.In Chapter 2,we illustrate the definition of rational Bézier curves/surfaces,NURBS curves/surfaces and the toric surface,together with their properties.We recall the toric degeneration of Bézier surface,which explains the geometric meaning of all of its weights tend to infinity.In Chapter 3,we present definitions of well-posed control polygons/control points sets of rational Bézier curves/surfaces.Based on toric degeneration of rational Bézier curves/surfaces,we present the sufficient and necessary conditions which imply rational Bézier curves/surfaces to be injective for any positive weights.We also present algorithms for the well-posedness iden-tification of the control polygons/control points sets.In Chapter 4,NURBS curves/surfaces can be transformed into rational Bézier curves/sur-faces after knot insertion.We define well-posed control polygons/control points sets of NURBS curves/surfaces.By using the toric degeneration and degree elevation of NURBS curves/sur-faces,the injectivity conditions of NURBS curves/surfaces with any positive weights are pre-sented.Chapter 5 concludes the whole thesis and presents the future work.