# Metrics on the circle

This article discusses various possible metrics that can be put on the circle.

## Subspace metric from Euclidean space

Under the standard embedding of the circle inside Euclidean space, we can give it the subspace metric. If the circle is embedded as having radius $r$, the subspace metric gives distance $2r$ to diametrically opposite points, and in general, for points separated by an angle of $\theta$, it gives the distance $2r \sin (\theta/2)$.

## Arclength metric

The arclength metric is the most natural metric on the circle. It arises as the metric from the Riemannian metric on the circle, and is a totally geodesic metric. The distance between two points is the length of the arc between them. Thus, if the circle is thought of as having radius $r$, the arclength metric gives a distance of $r\theta$ for points with an angular separation of $\theta$. Diametrically opposite points are separated by a distance of $\pi r$.