Analysis And Manipulation Of Singularity And Inflection Points On Parametric Curves

Geometric characteristics are one of important topics of study in the field ofCAGD. Singularity and inflection points of curves are important information to reflectgeometric characteristics of curves; the research of their existence has importanttheoretical significance and practical value. Traditional research methods of studygeometric characteristics mainly have algebraic and geometrical means. Although theyhave many advantages, there still exists insufficient, such as how to judge the shape ofcurves by the control polygons and manipulate singularity and inflection points ofcurves, etc.In order to overcome these shortcomings, this paper analyses and manipulates thesingularity and inflection points on some kinds of curves. On the one hand, thesingularity and inflection points on quadratic and cubic trigonometric spline curves andquasi quartic trigonometric spline curves lying in a class of control polygons areanalyzed, the necessary and sufficient conditions for these three kinds of curvesgenerating singularity and inflection points are obtained; On the other hand, themanipulation of singularity and inflection points on a class of parametric curves isinvestigated, some theorems and relevant conclusions for manipulating singularity andinflection points on the class of curves are obtained, the algorithm of controllingsingularity and inflection points of this class of curves is given, and application of thesetheorems and the algorithm are discussed. The main contents are organized as follows:In chapter one, the significance and development of the studies of singularity andinflection points of parametric curves are expounded. The studying methods ofsingularity and inflection points on parametric curves are summarized; the advantageand weakness of these methods are analyzed. Thus the studying problems of this paperare put forward, and the contents of this paper are summed up.In chapter two, an introduction to the definitions and properties of Bézier andC-Bézier curves, quadratic and cubic trigonometric spline curves, and quasi quartictrigonometric spline curves are given. In addition, a kind of quasi quartic trigonometricspline curves with two parameters is constructed, and its major properties are analyzed.Thus a solid foundation for the study work of following chapters is lied down. In chapter three and chapter four, the singularity and inflection points of quadraticand cubic trigonometric spline curves, and quasi quartic trigonometric spline curves areanalyzed. By studying the relation of a class of control polygons and these three kindsof curves, the necessary and sufficient conditions of singularity and inflection points areobtained respectively. The influence of shape parameters is discussed, the algorithm ofanalyzing singularity and inflection points and specific examples are also given.In chapter five, the manipulation of singularity and inflection points of a class ofparametric curves is investigated. Let a control point vary while rest n (n2)controlpoints fix, the singularity and inflection points can be generated and manipulated by themoving control point. It is obtained that singularity and inflection points belong to anarbitrary singular or inflection curve, and two arbitrary singular curves are tangent at thesingular point while two arbitrary inflection curves are concurrent at the inflection point.Moreover, the algorithm of controlling singularity and inflection points of the class ofparametric curves is given. And some specific examples of Bézier and C-Bézier curvesand quasi quartic trigonometric spline curves with two parameters are also given toshow the practicability and validity of the theorems and algorithm.In the last chapter, the summary of this paper is given and the problems for furtherstudy are put forward.