Definition:Spline
Definition
Draftsman's tool
A spline is a drawing tool which consists of a long strip of uniformly flexible material fixed in position at a number of points whose tension creates a smooth curve passing through those points.
Its purpose is to transferring that curve to another material.
Spline Function
Let $\closedint a b$ be a closed real interval.
Let $T : = \set {a = t_0, t_1, t_2, \ldots, t_{n - 1}, t_n = b}$ form a subdivision of $\closedint a b$.
Let $S: \closedint a b \to \R$ be a continuous function on $\closedint a b$ whose values on $t_0, t_1, \ldots, t_n$ are known.
On each of the intervals $\closedint {t_k} {t_{k + 1} }$, let $P_k: \closedint {t_k} {t_{k + 1} }: \R$ be a polynomial function such that:
- for $t$ on each of $t_k < t < t_{k + 1}$: $\map S t = \map {P_k} t$
The function $S: \closedint a b \to \R$ is known as a spline function on $T$.