α-曲线的构造及曲率单调条件的研究

The construction of α-curves and study of curvature monotony condition

在C-Bzier曲线的基础上提出了一种构造两条α-曲线的新方法,并分别给出了它们曲率单调的充分条件。研究结果表明类三次Bzier曲线和二次Bzier曲线分别是这两条α-曲线的特殊情况。此类α-曲线的特点是只有三个控制顶点且可通过改变形状参数来调整曲线的形状。前一条α-曲线起点处的曲率为零,可用一对α-曲线来构造两圆弧间半径比例不受限制的S型和C型G2连续过渡曲线;而单一的另一条α-曲线可用来构造两圆弧间不含曲率极值点的过渡曲线且当取特殊的形状参数时曲线终点处的曲率退化为零。最后,我们用实例表明了这两条α-曲线的有效性。

A new method to construct two types of α-curves based on C-Bzier curve is given.The sufficient conditions for α-curves of curvature monotony are deduced. The result shows that QuasiCubic Bzier curve and quadratic Bzier curve are special cases of α-curves. The advantage of α-curve is that it only has three control points and the shape of α-curve can be adjusted by a shape parameter,therefore α-curve is simpler and more powerful than C-Bzier curve. The first type α-curve has zero curvature at start point, and a pair of these α-curves can be used in constructing S-shaped or C-shaped G2 transition curves for separated circles, the ratio of two radii has no restriction. The second typeα-curve can be used in constructing a single transition curve with no curvature extreme for separated circles, and the endpoint curvature of this α-curve degenerate to zero when specific shape parameter is selected. Test examples are given to show the effectiveness of these two types of α-curves.