Spline

Given a pair of points and desired derivatives at each point, there is exactly one cubic between these points with these derivatives (note that there are precisely 4 constraints, and a cubic has 4 parameters). So we may build a spline between any n points if we specify the desired derivative at each point.

It turns out that a spline is a solution for a minimal energy problem for a flexible rod. This explains why splines look so good -- you actually see them in nature. It also explains the name -- Webster's 2^nd definition -- "a long, flexible piece of wood sometimes used as a ruler" -- was actually used by draughtsmen BC to draw smooth curves between control points. And since the piece of wood would assume the minimal energy configuration, they were in fact drawing splines.

Spline (?), n.

1.

A rectangular piece fitting grooves like key seats in a hub and a shaft, so that while the one may slide endwise on the other, both must revolve together; a feather; also, sometimes, a groove to receive such a rectangular piece.

2.

A long, flexble piece of wood sometimes used as a ruler.

© Webster 1913.