The following interpolation problem often arises in both engineering and mathematics: draw a smooth curve which passes through n + 1 points

in the plane. For instance, ship designers in the 19th century had to do this all the time. Draftsmen used, and still use, long thin elastic strips of wood or some other material to form a smooth curve passing through specified points. These strips, or splines, are held in place by lead weights called “ducks” at the specified points, see Fig. 28.1. For theoretical purposes we asume there is no friction between the spline and the ducks. The resulting curves usually have a pleasing appearance in addition to being smooth.

One way to find a smooth curve through the given points is to compute the shape of an elastic spline through them. However, this is quite complicated. Moreover, if the data points are scattered too much, frictionless ducks can not force a spline to go through them at all; the equations have no solution. This is a reflection of the physical situation. If we somehow force a spline to go through such a sequence of data points and then leave only frictionless ducks to hold it, the spline will balloon out between some ducks and no matter how long it is, it will pop out and get away from the ducks.

The Lagrange interpolation polynomials of Chapter 17 are simple to compute for any given sequence of points (28.1).