Constructing Of G~2 Quartic Bézier Interpolation Curve

In CAD/CAM modeling system,with the development of Surface modeling, more theories were proposed.Computer-Aided-Geometric-Design mainly researches on free curve and surface.And Bezier curve and surfaces are one kind of the most commonly used parametric curves in CAGD and Computer Graphics.It is an important problem to modify the shape of the Bezier curve and surface.Developing more convenient techniques for designing and modifying Bezier curve/surface is an important problem.When the Bezier curve/surface is generated,we always want to modify the shape of the curve/surface in order to satisfy the design.The solution of the practical problem often leads to solving optimization problem.According to the initial information,curves and surfaces modeling can be divided into two methods.One is free form designing,based on control points and parameters,designers can define curves or surfaces and modify interactively until the shapes satisfy the design objective.The other is the technology of interpolation and approximation.Curves and surfaces reconstructed from the given points by interpolation are called as interpolation curves and surfaces or by approximation called as approximation curves and surfaces.These two methods are all influenced by parameterization.So curves and surfaces parameterization and the technologies of interpolation and approximation are the foundation of geometric modeling and have important theory value and practical meaning.Besides,in the design of curves and surfaces one separate section of Bezier curves and surfaces has limited capacity,especially the various free curves and surfaces like automobile,airplane surface shapes,many sections are needed because of their complicated shapes and highly demanding smoothing.First of all a problem which users often encountered is that the extended curve unable to meet the accuracy requirements and it need to be re-adjusted.According to the issue,this paper presents a new method for constructing a G~2 quartic Bezier interpolation curve,referring to the curves' G~2 continuity and energy constraint to optimize the shape.According to the fact that geometric continuity in CAGD can accurately and objectively measure the degree of smooth for the connection between two spline parameter curves,by adding a control point and two degree of freedom on each sub-interval to construct a segment of the Bezier curve, we research the smooth connection between Bezier curves in G~2 continuity and proposes some examples.Compared with the existing methods of constructing G~2 curves,the curve by the new method has the properties that it has better flexibility and is locally adjustable.Moreover,G~2 curve by the new method offers additional degrees of freedom which can be used to increase the flexibility of construction and design to control the shape of the curve and therefore make the shape more desirable.