Enhetskulan

Enhetskulan, commonly denoted S^2 in three-dimensional Euclidean space, is the set of all points (x, y, z) with x^2 + y^2 + z^2 = 1. It is the boundary of the unit ball B^3 = {x^2 + y^2 + z^2 ≤ 1} and, as such, a 2-dimensional smooth manifold embedded in R^3. Equivalently, enhetskulan can be described as the space of all unit vectors in R^3, i.e., directions.

Geometrically, enhetskulan has constant Gaussian curvature 1 under the standard metric; its surface area is 4π. It is a compact, connected, orientable surface and is topologically equivalent to a sphere. The group of rotations SO(3) acts transitively on enhetskulan, making it a homogeneous space; every point can be rotated to any other.

Parametrizations: in spherical coordinates, x = sinφ cosθ, y = sinφ sinθ, z = cosφ with φ ∈ [0, π], θ ∈ [0, 2π). Geodesics on enhetskulan are great circles; the shortest path between two points is the arc of a great circle. In higher dimensions, the n-sphere S^n is defined similarly by x_1^2 + ... + x_{n+1}^2 = 1.

Applications and notes: the enhetskulan is widely used to represent directions, normal vectors, and orientations in geometry, physics, and computer graphics. In 2D, the unit circle is the analogous concept, sometimes specified as enhetscirkeln in Swedish; enhetskulan generally refers to the 2-sphere in three dimensions.