Metrics on the circle

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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.