Geometric Features Analysis Of A Class Of Cubic Parametric Curves In CAGD

Computer Aided Geometric Design (CAGD) plays an extremely important role both in Mathematics and Computer science and develops quickly. CAGD becomes a new cross-cover subject and it is close with every branch of mathematics and computer science. In these areas, the first thing is to solve the problem of design and manufacture of aviation, shipbuilding and industrial. Curve is the basic unit of the research in CAGD. These geometric features including the singularity, inflection point and convex distribution are an important research direction in geometric modeling. In this paper, the shape features of a class of Bezier curve with two shape parameters and a class of generalized Ball curve are analyzed by using the method based on the theory of envelope and topological mapping.In the first chapter, we introduce the development of the curves and surface with parameters CAGD in recent years. We also introduce Bezier curves with shape parameters, and Ball curve with a shape parameter and so on.In the second chapter, we introduce the research methods of the geometrical characteristics of the cubic Bezier curve and the cubic curve with parameters in the second chapter.In the third chapter, the shape features of a class of Bezier curve with two shape parameters are analyzed by using the method based on the theory of envelope and topological mapping. Investigate effects of the shape parameter on the curve shape. Necessary and sufficient conditions are derived for this curve having one or two inflection points, a loop or a cusp, or be locally or globally convex. Those conditions are completely characterized by the relative position of the edge vectors of the control polygon. Furthermore we discussed the influences of shape parameter on the shape diagram and the ability for adjusting the shape o f the curve.In the fourth chapter, we introduce a class of cubic generalized Ball curve. The Ball curve, cubic Bezier curve and the cubic Timmer curve are special case of the cubic generalized Ball curve. We analyze its geometric features.In the fifth chapter, we summarize the thesis. And the future research direction is prospected.