Research On The Theories And Methods Of Geometric Modeling Based On The Curves And Surfaces With Shape Parameter

This doctoral dissertation is devoted to curves and surfaces with shape parameter,including Bézier curves with shape parameter,triangular Bézier patches with shape parameter and its B-patch,Coons patches with shape parameter on triangles,as well as quadratic trigonometric spline curves with multiple-shape parameters.We give the construction method of these curves and surfaces,analyze the properties of their,such as continuity,approximability,convex—preserving and so on.Again we discuss the shape parameter how to influence the curves and surfaces.It is composed of six chapters.In chapter 1,we briefly introduce the historical background of computer aided geometric design and the history of the development of curves and surfaces in particular non-uniform rational B-spline,also describe some curves and surfaces different from NURBS.The main works of this paper are concluded as well.Chapter 2 is the extensions to Bézier curves.In which,we reform univariate Bernstein basis functions into a group of univariate blending functions,from which we define a class of new Bézier curves and surfaces with shape parameter.We analyze the impact of the shape parameter to the curves and surfaces of lower degree;also give the conditions between adjacent curves of lower degree and its application.Using tensor product, we extend the curves with shape parameter to the surfaces with shape parameter,and gives the conditions between adjacent surfaces of bi-four degree.In Chapter 3,we spread the discussion of triangular Bézier patches. Firstly,we extend the bivariate Bernstein basis functions to a group of bivariate blending functions,and then we define the triangular Bézier patches and B-patch with shape parameter.Furthermore,we show the properties of the patches,discuss the shape parameter how to impact the cubic and quartic patches and give the G~1-continuous conditions between adjacent cubic patches.Chapter 4 is the extensions of Coons patch on triangular domain.We discuss constructive problems of Coons patch on triangular domain in the C-type space,H-type space,T-type andλ-type space.Firstly,we give two classes of Hermite polynomials with shape parameter in different function space.Using projection operator and the method of convex combination,Coons patches with shape parameter on triangular domain are constructed.We can adjust the shape of Coons patches under boundary-valued shape-preserving.In Chapter 5,we give a new B-spline and Bézier Curves and Surfaces in the quadratic trigonometric polynomial space.Firstly,we define quadratic trigonometric polynomial basis functions with multiple shape parameters,then,we give quadratic trigonometric polynomial spline curves and Bézier curves.We show the properties of the curves,such as continuity,Approximability and so on,also discuss the impact of the shape parameter to the curves and the representation of ellipses(circle) by using these curves.Using tensor product,we extend the curves to the surfaces,and gives the conditions between adjacent trigonometric Bézier surfaces of bi-two degree.Chapter 6 is the conclusions of the paper and the study work to be carried out in the future.