Curvature Smoothing Of Combined Quadratic Generalized Bézier Curves

Computer Aided Geometric Design(CAGD)is an interdisciplinary subject that grew up with the development of modern industries such as shipbuilding and aircraft.The main research contents of CAGD are geometric representation and geometric modeling and fairing is one of the classical geometric problems.In product design,the smoothness of product shape is often required.There are a lot of algorithms for fairing curves and surfaces,but there is no unified definition for fairing.A generally accepted criterion of fair curve is that the curve contains relatively few curvature monotone segments.Therefore,it is necessary to study the monotonicity of curve curvature.Bézier curve with excellent properties is widely used in CAGD.With the development of q-calculus,Phillips q-Bézier curve and Lupas q-Bézier curve,two kinds of generalized Bézier curves containing qintegers,have also attracted much attention.In this paper,we will study the monotonicity of the curvature of the two kinds of generalized Bézier curves,the smooth blending of curves,and the construction of combined curves with monotonic curvature.On the one hand,for the quadratic Phillips q-Bézier curve,the plane rectangular coordinate system is established with the line between the first and last control vertices of the curve as the horizontal axis.The relationship between the curvature of the curve and the position of the middle control vertex is obtained,it is proved that the curvature has at most one extreme value and is a maximum.The necessary and sufficient conditions that the middle control vertex need to be satisfied when the curvature reaches the extreme value are obtained,and the geometric explanation is given.By defining the monotone curvature bounding circle,the necessary and sufficient conditions for monotone curvature are obtained,and the quadratic Phillips q-Bézier curve with monotone curvature is constructed based on the G2 blending conditions of two quadratic Phillips q-Bézier curves.Furthermore,a combined quadratic Phillips q-Bézier curve with monotonic curvature is constructed.The influences of parameters on the shape of blending curve are discussed,the similarities and differences between the G2 blending conditions of Bézier curve and that of quadratic Phillips q-Bézier curve are analyzed,and an example of transition curve with decreasing curvature is given.On the other hand,the monotonicity of the curvature of quadratic Lupas q-Bézier curve is discussed.Since the basis function of quadratic Lupas q-Bézier curve is a rational function,it is different from quadratic Phillips q-Bézier curve in the selection of coordinate system and the calculation of extreme points.By establishing an appropriate coordinate system,the necessary and sufficient conditions for the curve with curvature extreme and the necessary and sufficient conditions for monotonic curvature are obtained.When the middle control vertex of the curve lies on or within the bounding circle determined by the parameter q and the first and last control vertices,the curvature of the curve is monotonic.The influences of parameters in the G2 blending conditions of quadratic Lupas q-Bézier curve on the shape of splicing curve are analyzed.Furthermore,using the curvature monotone conditions and G2 blending condition,the combined quadratic Lupas q-Bézier curves with decreasing and increasing curvature are constructed respectively.When shape parameter q=1,quadratic Phillips q-Bézier curve and quadratic Lupas q-Bézier curve degenerate into classical quadratic Bézier curve.In this paper,the curvature monotone conditions of these two kinds of quadratic generalized Bézier curves are compared with those of classical quadratic Bézier curve.It is pointed out that when the curvature of the curve is monotonic,the selection range of middle control vertices of quadratic Lupas q-Bézier curve,classical quadratic Bézier curve and quadratic Phillips q-Bézier curve increases in turn.When the control vertices of the three curves are the same and meet their respective curvature monotony conditions,the shape of the quadratic Bézier curve is fixed,but by adjusting the q value,the shape of the quadratic Phillips q-Bézier curve can be farther away from the control polygon,and the shape of the quadratic Lupas q-Bézier curve can be closer to the control polygon.For the combined quadratic generalized Bézier curve with monotonic curvature,the shape of the curve can be adjusted appropriately by changing the variables in the condition of G2 blending and the shape parameter of the blending curve.