unitsphere

In mathematics, the unitsphere, also called the unit sphere, denoted S^{n-1}, is the set of points in Euclidean n-space with Euclidean norm equal to one: S^{n-1} = { x ∈ R^n : ||x|| = 1 }.

As a submanifold of R^n, it is smooth, compact, and of dimension n-1. For p ∈ S^{n-1}, the tangent space is T_p S^{n-1} = { v ∈ R^n : v · p = 0 }.

Its surface area is A(S^{n-1}) = 2π^{n/2} / Γ(n/2). For n=3, A = 4π. The geodesic distance on S^{n-1} between p and q is the arccos(p · q), equal to the angle between the two radius vectors.

Parametrizations: for S^{n-1}, one uses hyperspherical coordinates. For example, on S^2 in R^3, x = (cos θ sin φ, sin θ sin φ, cos φ) with θ ∈ [0, 2π) and φ ∈ [0, π].

Group of symmetries: rotations preserve the unit sphere; O(n) acts transitively. As a homogeneous space, S^{n-1} is isomorphic to SO(n)/SO(n-1).

Applications: the unitsphere serves as a natural domain for directional data and analysis, the normalization of vectors in machine learning and computer graphics, and as a constraint or target manifold in optimization and geometry. It also provides foundational examples in differential geometry and topology.