Hypersphere

A hypersphere, in n-dimensional Euclidean space, is the set of all points at a fixed distance r from a given center. In R^d with center c, the d-dimensional hypersphere (the surface) is {x ∈ R^d : ||x − c|| = r}. The solid ball of radius r is {x ∈ R^d : ||x − c|| ≤ r}. In many contexts the term hypersphere refers specifically to the surface, while the solid is called a ball. The hypersphere is a smooth (d−1)-dimensional manifold with full rotational symmetry described by the orthogonal group O(d).

Equation in coordinates: sum_{i=1}^d (x_i − c_i)^2 = r^2.

Volumes and areas: The d-dimensional ball has volume V_d(r) = π^{d/2} r^d / Γ(d/2 + 1). The boundary, the (d−1)-dimensional hypersphere, has surface area A_{d−1}(r) = 2 π^{d/2} r^{d−1} / Γ(d/2). Equivalently, A = d V_d(r)/r.

Special cases: In d=2 the hypersphere is a circle with circumference 2πr and area πr^2; in d=3 it is the ordinary sphere with surface area 4πr^2 and volume (4/3)πr^3.

Coordinate systems: Hyperspherical coordinates generalize polar coordinates, with a radial coordinate r and d−1 angular coordinates, useful for integration on S^{d−1}.

Generalizations: In topology, S^n denotes the unit n-sphere, i.e., the set of points in R^{n+1} with ||x|| = 1. The word hypersphere is often used interchangeably for the boundary of a ball in any dimension.