Identification Of Singularities And Inflection Points On Parametric Curves

This thesis mainly makes some research on the existence of singularities and inflection points on some planar and space curves whose these special points can be generated or manipulated. It includes the discussion about singularities and inflection points on rational parametric curves, C-curves and C-Bézier curve especially, and F-Bézier curve.Firstly, we briefly review some research related on singularities and inflection points on some parametric curves in this thesis. These curves include rational curves, planar rational cubic curves, rational Bézier curves, planar C-curves, planar C-Bézier curve etc.Secondly, we on one hand use envelope theory and topological wapping method which given by Ye Zhenglin and Wu Rongjun to obtain the distribution graphs about the singularities and inflection points of planar F-Bézier curves, which show their existence and convexity. On the other hand, we discussion the problem that how to generate or manipulate these special points. While three control points are fixed, these points can be generated or manipulated by their fourth control points(mainly end points). It is proved that the singular point or inflection point belongs to two singular curves or two inflection curves which are generated by the first 3 and the last 3 control points respectively. And two singular curves are tangent at the singular point. Therefore, we can know not only how to identify the existence of singularities and inflection points, but also how to manipulate these points according to the practial application.