目的 为了使得过渡曲线的设计更为简单高效。提出基于3个控制顶点的类三次Bézier螺线。方法 通过对基函数的研究首先构造了3条在一定条件下曲率单调递减的类三次Bézier曲线,并由参数的对称性得另3条曲率单调递增的曲线。它们具有端点性、凸包性、几何不变性等三次Bézier曲线的基本性质,特点是只有3个控制顶点。接着严格地证明了此类曲线曲率单调的充分条件。 结果 有两条曲线比三次Bézier曲线的曲率单调条件范围大,且类三次Bézier螺线与三次Bézier螺线存在一定的位置关系。这6条曲线中有4条曲线的一个端点处曲率为零,可组合成4对类三次Bézier螺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;剩下的两条曲线在两圆弧半径相差较大的情况下都可做不含曲率极值点的过渡曲线。最后用实例表明了此类曲线的有效性。结论 在过渡曲线设计中基于3个控制顶点的类三次Bézier螺线比三次Bézier螺线更为简单高效。

Objective This study aims to design a simple and effective transition curve. Method First, three quasi-cubic Bézier curves with monotone decreasing curvature are constructed after studying the base functions. Three other quasi-cubic Bézier curves with monotone increasing curvature are then obtained by parameter symmetry. These new curves have similar properties with the cubic Bézier curves, including endpoint, convex hull, and geometry invariability properties. However, unlike the cubic Bézier curves, the quasi-cubic Bézier curves only have three control points. We then provide strict mathematical proofs in relation to the sufficient conditions of the monotone curvature of these curves. Result Two of the quasi-cubic Bézier curves covered a broader scope than the cubic Bézier curves with monotone curvature. The specific positional relationship between the quasi-cubic Bézier spiral curves and the cubic Bézier spiral curves were determined. Four of the six quasi-cubic Bézier curves had zero curvature at one endpoint and can therefore be combined into four pairs of S-shaped or C-shaped transition curves for separated circles; the ratio of two radii had no restriction. The remaining two quasi-cubic Bézier curves can be used to form a single transition curve with no curvature extreme for separated circles when the difference of two radii is large. Finally, examples were given to show the effectiveness of these curves. Conclusion In transition curve design, the quasi-cubic Bézier spiral curve with three control points is more simple and effective than the cubic Bézier spiral curve.