Research On The C-Bézier Curve And Surface With Shape Parameters

With the increasing demand of product design,many products need to design the geometric shape of curve and surface before manufacturing,such as the car shell design,the design of aircraft wing,and the shoes and clothes people wear in daily life.These are all the research of curve and surface modeling.The research of curve and surface modeling has always been the core content of CAGD.In addition,it is very important to study the curve and surface which satisfiy some properties by interpolation,fitting,approximation and splicing.In the modeling of surface,C-Bézier model has been widely used in the design of free-form curved surface in recent 20 years.The C-Bézier model not only retains the excellent characteristics of the Bézier model,but also the shape adjustability is the most prominent feature of the C-Bézier model.Therefore,the C-Bézier model is widely used in curve and surface modeling,which means that it has practical and theoretical value for its research.The traditional Bézier developable surface can only control the shape of the surface by controlling points,which makes the shape control ability greatly weakened.Of course,the rational Bézier developable surface can be used to control the shape by its weight factor,but this construction method also has its own shortcomings.One is to change the weight factor of the rational developable surface,and all the rational Bézier basis functions will change.The other is that the rational curve and surface are very complex in the derivation and integration.Therefore,it is necessary to find a method to construct developable surface with easy shape manipulation.In addition,C-Bézier curve and surface with shape parameters are more flexible and convenint to change their shape without changing their control points.Therefore,a method of interpolating geodesic and curvature lines to construct the cubic C-Bézier developable surface with shape parameters is proposed.The conditions of parameter continuity and geometric continuity of C-Bézier curves with shape parameters are also studied.The main research contents are as follows:1.Geodesic and curvature lines are a kind of special curves on the surface.Because of their unique geometric characteristics,they play an important role in the analysis and description of the surface.In this paper,a method of constructing cubic C-Bézier developable surface with shape parameters by interpolating geodesic and curvature lines is proposed.The concrete expression of the surface is given.The necessary and sufficient conditions for the developable surface to be cylinder or cone are also analyzed.The main research content is to introduce the C-Bézier basis functions with shape parameters,and use them to construct cubic C-Bézier curves.Finally,the necessary and sufficient conditions for the given curves to be geodesic and curvature lines on the constructed C-Bézier developable surface are derived.2.In CAD / CAM,we often encounter the modeling problem of complex curves,and complex curves are often difficult to be represented by a single curve,so how to realize the curve splicing and make them convenient and flexible to be applied to all kinds of curve and surface modeling is the problem we need to solve.In this paper,the conditions of parametric and geometric continuity of C-Bézier curve with shape parameters are studied.This paper mainly studies the smooth stitching of cubic C-Bézier curve with shape parameters,and gives the relationship equations of the control vertices.Due to the existence of continuity conditions and different shape parameters,it is more convenient and flexible to change the shape of the curve without changing its control points.