摘要
In parametric interpolation, an ordered set of n points $\{ P_i :(x_i ,y_i )\} ,i = 0,1, \cdots ,n - 1$, is given, and a type of interpolating function is specified. The points are then parametrized; that is, for each i, a value $t_i $ is assigned to the point $P_i $. It is then required to use the n pairs $\{ (t_i ,x_i )\} $ and the n pairs $\{ (t_i ,y_i )\} $ separately to obtain interpolating functions \[ (1)\qquad x = X(t),\qquad y = Y(t) \] of the specified type. Equations (1) then define a curve parametrically which interpolates the given points.
It is obvious that the $t_i $ must be distinct and monotonic with i, but there appears to have been little analysis devoted to the influence of the choices of the $t_i $ on the subsequent curve. There is an intuitive feeling, however, that some concept of “distance between the points” should govern the parametrization.
In this paper, we consider closed curves which are parametric, periodic, cubic splines and show that the parametrization can influence the existence of singularities. Specifically, if the parametrization is based on a vector-norm distance, the curve cannot have a corner, while if the parameters are assigned as consecutive integers (as in some computer routines), a corner can occur even though the defining equations would appear to make that impossible.
摘要译文
在参数插值，n个点$ \\有序集合{P_i：（x_i，y_i）\\}，I = 0,1，\\ cdots，正 - 1 $，给出，和类型的内插函数被指定。然后这些点参数化;也就是说，对于每个i，一个价值$ t_i $被分配给点$ P_i $。使用N对$ \\ {（t_i，x_i）\\} $和N对$ \\ {（t_i，y_i）\\} $分别获得插值函数\\ [（1）\\ qquad X = X是则需要（T），\\ qquad Y = Y（T）\\指定类型的。等式（1）然后定义一个曲线参插该给定点。\r
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很明显，在$ t_i $必须是不同的和单调的，其中i，但一直很少分析倾注到后续曲线上的$ t_i $的选择的影响出现。有一个直观的感受，但是，“两点之间的距离”的概念，一些应指导的参数化。\r
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在本文中，我们考虑封闭曲线，该曲线是参数，周期性的，三次样条和表明该参数化可以影响奇点的存在。具体地说，如果参数化是基于矢量的范数距离，曲线不能有一个角落，而如果参数被指定为连续的整数（如在某些计算机程序），一个角可以即使限定方程似乎发生作出这样的不可能。
M. P. Epstein. On the Influence of Parametrization in Parametric Interpolation[J]. SIAM Journal on Numerical Analysis, 1976,13(2): 261-268