On the Influence of Parametrization in Parametric Interpolation

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 exi...