---
tag: Z2Z9BE946H
kind: definition
name: Osculating circle
msc: [53A04]
defines: NWSFMCRWHJ
uses: [7JQS6VKFEZ, 06JMR2XWXW, 0D98771SM0, 045DN06F3S]
sources:
  - url: https://en.wikipedia.org/wiki/Osculating_circle
---

## Definition

Let $\gamma \colon I \to \R^2$ be a [[regular|06JMR2XWXW]] curve of class $C^2$ on an interval $I$, and let $t_0 \in I$.
Let $T$ be its unit [[tangent vector|0D98771SM0]], let $N$ be $T$ rotated by a right angle counterclockwise, and let $k$ be its signed [[curvature|7JQS6VKFEZ]].
Assume that $k(t_0) \ne 0$.
The *centre of curvature* at $t_0$ is $c = \gamma(t_0) + N(t_0) / k(t_0)$, and the *radius of curvature* is $1 / \abs{k(t_0)}$.
The *osculating circle* of $\gamma$ at $t_0$ is the [[circle|045DN06F3S]] with centre $c$ and radius $1 / \abs{k(t_0)}$.

## Remarks

Example: for $\gamma(t) = (t, t^2)$ at $t_0 = 0$, one has $T = (1, 0)$, $N = (0, 1)$ and $k(0) = 2$.
So the osculating circle has centre $(0, 1/2)$ and radius $1/2$.
Non-example: for $\gamma(t) = (t, t^3)$ at $t_0 = 0$, the curvature is $0$, so there is no osculating circle there; the tangent line plays its role.
The osculating circle passes through $\gamma(t_0)$, has the same tangent there, and is the only circle with contact of order at least $2$ with the curve.
For a curve in $\R^3$ with $\kappa(t_0) > 0$, the osculating circle is defined in the same way, with $N$ the principal normal.
It lies in the osculating plane, spanned by $T$ and $N$.