曲率单调的组合二次Phillips q-Bézier曲线

Combinatorial quadratic Phillips q-Bézier curves with monotone curvature

Phillips q-Bézier曲线是一类包含q-整数的广义Bézier曲线。针对二次Phillips q-Bézier曲线的曲率单调条件,从代数和几何两方面进行了研究,构造出曲率单调的二次Phillips q-Bézier曲线及曲率单调递减的组合二次Phillipsq-Bézier曲线。首先,通过曲线曲率的坐标表示,探究代数形式的曲率单调条件,定义曲率单调包围圆,给出二次Phillips q-Bézier曲线具有单调曲率的几何充要条件。当形状参数q=1时,Phillips q-Bézier曲线退化为经典的Bézier曲线,因此上述曲率单调条件包含经典二次Bézier曲线的结果。其次,讨论二次Phillips q-Bézier曲线间的G^(2)光滑拼接条件及条件中的各个参数对拼接曲线的影响。再次,对于给定首末控制顶点的曲线,选择合适的中间控制顶点,求得使其具有单调曲率时形状参数的取值范围,构造出曲率单调的单条二次Phillips q-Bézier曲线。进而,构造出同时满足G^(2)拼接与曲率单调递减的组合二次Phillips q-Bézier曲线。最后,利用曲率单调递减的组合二次Phillipsq-Bézier曲线,构造出具有包含关系的两圆之间的缓和曲线。数值实例显示了组合二次Phillips q-Bézier曲线的造型优势和灵活性。

Phillips q-Bézier curves are a class of generalized Bézier curves containing q-integers.The research was conducted on the curvature monotonicity condition of quadratic Phillips q-Bézier curve from two aspects of algebra and geometry.Based on this,the following two curves were constructed:a quadratic Phillips q-Bézier curve with monotonous curvature and a combined quadratic Phillips q-Bézier curve with decreasing curvature.Firstly,through the coordinate representation of curve curvature,this paper explored the condition of monotonic curvature in algebraic form.By defining the curvature decreasing(or increasing)bounding circle,the geometric sufficient and necessary conditions were given to enable decreasing(or increasing)curvature for quadratic Phillips q-Bézier curves.In the case of the shape parameter q=1,Phillips q-Bézier curves would degenerate into classical Bézier curves.Thus,the curvature monotonicity conditions of quadratic Phillips q-Bézier curves include the results of classical quadratic Bézier curves.Secondly,the paper examined the G^(2)smooth condition of quadratic Phillips q-Bézier curves and the influence of parameters on the stitching curve.Thirdly,for the quadratic Phillips q-Bézier curve with given initial and final control vertices,the appropriate intermediate control vertex was selected,the range of shape parameters was obtained in the case of decreasing(or increasing)curvature,and a quadratic Phillips q-Bézier curve with decreasing(or increasing)curvature was constructed.Furthermore,a combined quadratic Phillips q-Bézier curve was constructed,which could satisfy both G^(2)smooth condition and decreasing curvature.Finally,using the combined quadratic Phillips q-Bézier curve with decreasing curvature,the transition curve between two circles with inclusion relationship was constructed.The numerical examples highlight the advantages and flexibility of the combinatorial quadratic Phillips q-Bézier curve in modeling.