Spline approximation of planar discrete data

缓和曲线有几个优点，既不包含拐点，也不包含奇点，也不包含曲率极值。在设计视觉上令人满意的曲线时，这种曲线对于现有曲线的延伸和现有曲线之间的过渡非常有用。这种由三次样条曲线组成的三次螺线，即仅在指定位置具有曲率极值的曲率连续曲线，在公路或铁路路线的设计、移动机器人的轨迹设计或数控切割机械的切割路径设计中都是理想的应用。首先，我们得到了在给定数据的每个子区间上可以用T-三次或圆弧“/T-三次逼近控制点上的单位切向量的条件；其次，我们给出了一种平面三次样条线等距曲线的三次样条逼近算法，并给出了三次逼近法的一个易于计算的算法和存在等距长度的一个充分条件；第三，我们证明了，如果光滑螺线上的两个插值点足够近，则插值到G^2 Hermite数据的两点三次样条线也是螺线的。

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.