nLab
unit sphere

Idea

Given a metric space $(X,d)$ and a point $x \in X$, then the unit sphere $S_x(X) \subset X$ is the subset of those points with unit distance from $x$:

$$S_x(X) \;\coloneqq\; \left\{ x' \in X \;\vert\; d(x',x) = 1 \right\} \,.$$

Examples

In the Euclidean space $(X,d) = E^n$ of dimension $n$, the unit sphere is the usual (n-1)-sphere $S^{n-1} \simeq S_0(\mathbb{R}^n)$. For $n = 2$ this is the unit circle, for $n = 3$ the unit 2-sphere and so on.

Related concepts

• stereographic projection

• unit sphere bundle

• direction of a vector

• polar coordinates