unit sphere (changes) in nLab

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Contents

Idea

Given a metric space (X,d)(X,d) and a point x∈Xx \in X, then the unit sphere S x(X)⊂XS_x(X) \subset X is the subset of those points with unit distance from xx:

S x(X)≔{x′∈X|d(x′,x)=1}. S_x(X) \;\coloneqq\; \left\{ x' \in X \;\vert\; d(x',x) = 1 \right\} \,.

Examples

In the Euclidean space (X,d)=E n(X,d) = E^n of dimension nn, the unit sphere is the usual (n-1)-sphere S n−1≃S 0(ℝ n)S^{n-1} \simeq S_0(\mathbb{R}^n). For n=2n = 2 this is the unit circle, for n=3n = 3 the unit 2-sphere and so on.

• stereographic projection

• unit sphere bundle

• direction of a vector

• polar coordinates