Unit sphere
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In mathematics, a unit sphere is the set of points of distance 1 from a fixed central point, where a generalized concept of distance may be used; a unit ball is the set of points of distance less than 1 from a fixed central point. Usually a specific point has been distinguished as the origin of the space under study and it is understood that a unit sphere or unit ball is centered at that point. Therefore one speaks of "the" unit ball or "the" unit sphere.
A unit sphere is simply a sphere of radius one. The importance of the unit sphere is that any sphere can be transformed to a unit sphere by a combination of translation and scaling. In this way the properties of spheres in general can be reduced to the study of the unit sphere.
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[edit] Unit spheres and balls in Euclidean space
In Euclidean space of n dimensions, the unit sphere is the set of all points 
which satisfy the equation
and the closed unit ball is the set of all points satisfying the inequality
[edit] General area and volume formulas
The volume of the unit ball in n-dimensional Euclidean space, and the surface area of the unit sphere, appear in many important formulas of analysis. The surface area of the unit sphere in n dimensions, often denoted ωn in the literature, can be expressed by making use of the Gamma function. It is
.
The volume of the unit ball is ωn / n.
[edit] Non-general area and volume formulas
In three-dimensional Euclidean space, a unit sphere's volume is
and its surface area is
- A = 4π.
[edit] Unit balls in normed vector spaces
More precisely, the open unit ball in a normed vector space V, with the norm 
, is
.
It is the interior of the closed unit ball of (V,||·||),
.
The latter is the disjoint union of the former and their common border, the unit sphere of (V,||·||),
.
[edit] Comments
The 'shape' of the unit ball is entirely dependent on the chosen norm; it may well have 'corners', and for example may look like [−1,1]n, in the case of the norm l∞ in Rn. The round ball is understood as the usual Hilbert space norm, based in the finite dimensional case on the Euclidean distance; its boundary is what is usually meant by the unit sphere.
[edit] Generalizations:
[edit] Metric spaces
All three of the above definitions can be straightforwardly generalized to a metric space, with respect to a chosen origin. However, topological considerations (interior, closure, border) need not apply in the same way (e.g., in ultrametric spaces, all of the three are simultaneously open and closed sets), and the unit sphere may even be empty in some metric spaces.
[edit] Quadratic forms
If V is a linear space with a real quadratic form F:V → R, then { x ∈ V : F(x) = 1 } is sometimes called the unit sphere of V. Two-dimensional examples occur with split-complex numbers and dual numbers. When F takes negative values, then {x ∈ V: F(x) = − 1} is called the counter-sphere.
