---
tag: 84TJ8KQ926
kind: concept
name: Spline
msc: [41A15, 65D07]
sources:
  - url: https://en.wikipedia.org/wiki/Spline_(mathematics)
same_as:
  wikidata: Q735256
  wikipedia: Spline (mathematics)
---

## Summary

A spline is a function made of polynomial pieces on consecutive intervals.
The pieces are joined at points called knots, so that the whole function is as smooth as the pieces allow.
A spline of degree $n$ is usually required to have $n - 1$ continuous derivatives.
Splines combine the flexibility of piecewise functions with the simplicity of polynomials, and they avoid the oscillations of high-degree polynomial interpolation.
They are central in [[spline interpolation|D59HBKKRP9]], in approximation theory and in computer-aided design, where [[B-splines|62AFX32JGT]] give a convenient basis.

## Explanations

### L1 human

The name comes from the thin flexible strips of wood that draftsmen bent through fixed pegs to draw smooth curves.
A mathematical spline does the same with polynomial pieces: each piece is simple, and the pieces meet without corners or jumps in bending.
The simplest example is a broken line, a spline of degree $1$: straight pieces joined continuously.
Cubic splines use pieces of degree $3$ joined so that position, slope and curvature all match.
The [[definition|KTVJR95GSF]] makes the conditions precise.