On Degenerations Of NURBS Curves And Surfaces

Curve and surface modeling,which focuses on how to express,design and process curve and surface in computer system,plays an important role in Computational Geometry,Computer Aided Geometric Design(CAGD)and Computer Aided Design(CAD).Bezier method,B-spline method,and non-uniform rational B-spline(NURBS)method(which is the generalization of Bezier method and B-spline method)are the common representations of curves and surfaces.NURBS method is also a mathematical model commonly used in Computer Aided Design and Manufacturing(CAD/CAM).It can represent not only standard analytic surfaces,such as conical surface,surface of revolution and general quadric surface,but also complex freeform surfaces.Due to these advantages,NURBS is the only mathematical method to express the geometric shapes of industrial products by International Standard Organization(ISO).Usually,the shape modification of NURBS curves and surfaces can be achieved by means of changing knot vectors,control points,and weights.Similar to the geometric meaning of the weight of rational Bezier curve and surface,when a single weight approaches infinity,the limit of a NURBS curve or surface tends to the corresponding control point.However,when all weights approach infinity,the geometric structure and properties of the degeneration form(or limit form)of NURBS curve and surface are still poorly studied.The toric surface is a kind of multi-sided rational parametric surface,whose mathematical theory is based on toric variety from algebraic geometry and toric ideals from combinatorics.The toric surface preserves most properties of rational Be6zier surfaces,since its basis functions are the generalization of classical Bernstein basis functions.The toric surface approaches its regular control surface when all weights tend to infinity,which is called the toric degenerations of the toric surface.In this thesis,based on the toric degenerations of toric surface,we present the limit properties of NURBS curves and surfaces when all weights tend to infinity in the form of power function,i.e.,the toric degenerations of NURBS curves and surfaces.Our results extend the understanding of the geometric meaning of weights of NURBS curves and surfaces.We also study the degenerations of rational Bezier curves and surfaces and NURBS curves and surfaces when all weights tend to infinity in the form of exponential function.Moreover,the application of degeneration theory in computer animation,shape modification,deformation of curves and surfaces are also explored.The main contents of this thesis can be summarized as follows:1.We study the toric degenerations of NURBS curve/surface by using the toric degener-ations of toric surface presented by Garcia-Puente et al.We define the regular control curve/surface,which is a kind of control structure of NURBS curve/surface,and prove that it is exactly the limiting position of the NURBS curve/surface when all weights tend to infinity in the form of power function.This result not only explains the geometric structure of the limit curve/surface when all weights of NURBS curve/surface tend to in-finity,but also extends the understanding of the geometric meaning of weights of NURBS curve/surface.2.We study the degenerations of the rational Bezier curves and surfaces with weights in the exponential function by defining the rational Bezier curves and surfaces with weights in the exponential function.Based on the toric degenerations of toric surface,we analyze re-lations of our result and the work of the toric degenerations of rational B6zier curves and surfaces presented by Garcia-Puente et al.We also illustrate the geometric structure and properties of the rational Bezier curves and surfaces with weights in the exponential func-tion.The work of this thesis generalizes the geometric properties of weights of rational Bezier curves and surfaces.3.We study the degenerations of the NURBS curves and surface with weights in the expo-nential function by defining the NURBS curves and surface with weights in the exponen-tial function.Based on the toric degenerations of toric surface,we analyze relations of our result and the work of the toric degenerations of NURBS curves and surfaces.We also illustrate the geometric structure and properties of the NURBS curves and surfaces with weights in the exponential function.The work of this thesis generalizes the geometric properties of weights of NURBS curves and surfaces.