2. Interpolation

Interpolation for parametric curves means finding a curve $P (t)$ that goes through given data points at given times, as in Figure .

In other words, data consists of points $P_0,\dots, P_n$ and times $t_0,\dots, t_n$. The problem is to find a parametric curve $P (t)$ such that $P(t_k) = P_k$ for each $k$. As always, let's assume we're working in    R$^2$, although everything works similarly in    R$^n$ for any $n$.

Figure: An interpolated curve

book/07dir/general.eps

But what kind of curve? There are actually several good possibilities, but one of the simplest is to have $P (t)$ be a polynomial curve. What degree should we hope for? Because two points determine a line and three a parabola, it seems reasonable to ask that for $n+1$ data points $P_0,\dots, P_n$, the degree of the curve should be at most $n$. This is always possible:

Theorem. 2.1 (Lagrange). For data points $P_0,\dots, P_n$ and times $t_0,\dots, t_n$, there is a unique polynomial curve $P (t)$, of degree at most $n$, such that $P (t_0)$ $=$ $P_0,\dots, P (t_n)$ $=$ $P_n$, provided only that $t_0,\dots, t_n$ are distinct.

A problem fitting this theorem can be called a Lagrange interpolation problem.