Research On Bézier Curves And Surfaces With Shape Parameter

This thesis summaries our researches on the parameter curves and surfaces modeling—the trigonometric polynomial T-Bézier curves with shape parameter and the hyperbolic polynomial H-Bézier curves with shape parameter.At first,we review the history of computer aided geometric design,especially free form curves and surfaces.Then we present the Bézier curves in brief and the research background of article,narrating Bézier curves with shape parameter and trigonometric and hyperbolic polynomial C-Bézier,remarking on their excellence and disadvantage.Afterward,we generate two initial functions on trigonometric polynomial space and hyperbolic polynomial space.The basis functions of nth order trigonometric polynomial T-Bézier curves with shape parameter hyperbolic polynomial H-Bézier curves with shape parameter are constructed by an integral approach.Two new basis functions provides properties analogous to Bernstein basis.Based on these basis functions,we define the T-Bézier curves with shape parameter and the H-Bézier curves with shape parameter.Then the shape parameter can adjust the curves'shape with the same control polygon.As the increase of the parameter,the T-Bézier curves with shape parameter and the H-Bézier curves with shape parameter approximate to the control polygon.,The T-Bézier curves with shape parameter and the H-Bézier curves with shape parameter have many properties analogous to Bézier curves,and they can represent transcend curves,such as catenary,hyperbola and helix.In addition,this thesis gain the elevation formulae of the T-Bézier curves with shape parameter and the H-Bézier curves with shape parameter.According to the elevation formulae,we prove the convergence of control polygons by elevation.At the same time we research on C~1,C~2 continuous condition,so the shape parameter can have local modification and holistic modification on connecting two curves.Furthermore,we generalize these two curves to the corresponding tensor product surfaces.Shape parameter can adjust surface well.These researches are important in the computer aided geometric design.We believe that new Bézier curves with shape parameter given will be a new powerful tool for freeform curves and surfaces.