---
tag: M39DK862R7
kind: definition
name: Curvature of a curve
msc: [53A04]
defines: 7JQS6VKFEZ
uses: [C57T562NKB, 06JMR2XWXW, 0D98771SM0, 00R1JDDV5X, 0SEDGXG9S3]
sources:
  - url: https://en.wikipedia.org/wiki/Curvature
---

## Definition

Let $n \ge 2$, let $I \subseteq \R$ be an interval, and let $\gamma \colon I \to \R^n$ be a [[regular|06JMR2XWXW]] [[differentiable curve|C57T562NKB]] of class $C^2$.
Let $T(t) = \gamma'(t) / \norm{\gamma'(t)}$ be its unit [[tangent vector|0D98771SM0]].
The *curvature* of $\gamma$ at $t \in I$ is
$$\kappa(t) = \frac{\norm{T'(t)}}{\norm{\gamma'(t)}}.$$
If $\gamma$ is parametrised by [[arc length|00R1JDDV5X]] $s$, this reads $\kappa(s) = \norm{\gamma''(s)}$.

## Remarks

Example: the circle $\gamma(t) = (r \cos t, r \sin t)$ with $r > 0$ has $\norm{\gamma'(t)} = r$ and $T(t) = (-\sin t, \cos t)$, so $\norm{T'(t)} = 1$ and $\kappa(t) = 1/r$.
Non-example: for $\gamma(t) = (t^2, t^3)$ the formula is not defined at $t = 0$, because $\gamma'(0) = 0$; the curve is not regular there.
The curvature does not depend on the regular parametrisation chosen.
In $\R^3$ it is given by $\kappa = \norm{\gamma' \times \gamma''} / \norm{\gamma'}^3$, where $\times$ is the [[cross product|0SEDGXG9S3]].
For a plane curve $\gamma = (x, y)$ one also uses the *signed curvature*
$$k = \frac{x' y'' - y' x''}{(x'^2 + y'^2)^{3/2}},$$
which is positive where the curve turns left; then $\kappa = \abs{k}$.