An Extension Of Bézier Curve And Its Application

The representation method of curve and surface are an important part of the study ofComputer Aided Geometric Design, and the choice of basis function has an importantinfluence on the nature of the curve and surface. The Bernstein polynomial is basisfunction of the Bézier curves and the Bézier surfaces, which has been widely used inCAGD fields. But for given control points, the Bezier curve is uniquely identified. Howto control the shape of curves under the condition of control points are unchanged is animportant problem that we often encountered in the design of curves and surfaces.Rational Bézier method can control the shape of a rational Bezier curve or surface byusing weights, but the weights effect on the shape of curves and surfaces is not obvious,and the calculation of its derivation and integration are relatively complex. In order tomake the Bezier curve is adjustable, people began to promote the Bezier curve, and aseries of fruitful results have been achieved. In this paper, we mainly studied the allkinds of existing Bezier curves with shape parameter. On this basis a new extensionmethod of nth degree Bezier curve is proposed. And we discussed the continuousstitching condition and application of extending curves.This paper includes six chapters altogether:The first chapter mainly introduced the development history and the background ofcurve and surface modeling, and the current research status of extension curves. Thesecond chapter firstly introduced the definition and the nature of Bézier curves, and thenpresented the curve continuous stitching conditions. The third chapter firstly introducedthree different expanding types of the basis functions of a5th degree Bézier curve, thensummarized the theoretical proof about expanding types of4th degree Bézier curve,finally listed the expression of basis functions of Q-Bézier curve in the existingliterature. In the fourth chapter a new extension method of nth degree Bezier curve isproposed. A class of blending function of degree n+1(n≥2) with shape parameter λ ispresented. The classical Bernstein basis function is a special case of the above functions.Based on the blending function, a method of generating piecewise polynomial curveswith a shape parameter is given. The properties of the curves and its blending functionsare analyzed. The fifth chapter discussed the G2continuity condition and someapplication examples of extending curves, and then simply introduced extension Béziersurfaces. The sixth chapter summarized the full text and put forward the future work.