---
tag: NWSFMCRWHJ
kind: concept
name: Osculating circle
msc: [53A04]
sources:
  - url: https://en.wikipedia.org/wiki/Osculating_circle
same_as:
  wikidata: Q2166240
  wikipedia: Osculating circle
---

## Summary

The osculating circle of a curve at a point is the circle that fits the curve best at that point.
It touches the curve there, with the same tangent line and the same [[curvature|7JQS6VKFEZ]].
Its radius is the radius of curvature $1/\kappa$, and its centre is the centre of curvature.

## Explanations

### L1 human

Stand at a point of a curved road and try to draw a circle that follows the road as long as possible near you.
On a gentle bend the best circle is huge, and on a sharp bend it is small.
That best circle is the osculating circle; the word comes from the Latin for "kissing".
For the parabola $y = x^2$ at the origin, the osculating circle has radius $1/2$ and centre $(0, 1/2)$.