On the Influence of Parametrization in Parametric Interpolation

Abstract

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.