Spline approximation of planar discrete data

螺旋具有不含拐点、奇点和曲率极值的优点。这样的曲线对于现有曲线的延伸和现有曲线之间的过渡设计非常有用。这种由三次样条组成的三次螺旋，即仅在指定位置具有曲率极值的曲率连续曲线，对于公路或铁路路线或移动机器人轨迹或数控切割机械切割路径的设计等应用是理想的。首先，我们得到了在给定数据的每个子区间上是否可以使用t -三次或arc”/ t -三次的控制点上单位切向量的条件。其次，我们提出了平面三次样条偏移曲线的三次样条逼近算法，并推导了三次逼近方法的一种易于计算的算法和偏移长度存在的充分条件。第三，我们证明了两点三次样条插值到从螺旋上取的G^2 Hermite数据上，如果平滑螺旋上的两个插值点足够接近，也是螺旋。

Spirals have several advantages of containing neither inflection points, singularities nor curvature extrema. Such curves are useful for extension of an existing curve and transition between existing ones in the design of visually pleasing curves. Such cubic spirals composed of cubic splines, i.e. , curvature continuous curves with curvature extrema only at specified locations are desirable for applications as the design of highway or railway routes or the trajectories of mobile robots or the cutting paths for numerically controlled cutting machinery. First we have obtained the condition on the unit tangent vectors at the controlled points if T-cubic or arc"/ T-cubic can be used on each subintervals for the given data. Secondly, we have proposed an algorithm for a cubic spline approximation of an offset curve for a planar cubic spline and derived an easier to calculate algorithm for the cubic approximation method and a sufficient condition on an offset length for the existence. Thirdly, we have shown that that two-point cubic splines interpolating to the G^2 Hermite data taken from the spiral are also spirals if the two interpolation points on the smooth spiral are close enough.