 Unit sphere

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 closed unit ball is the set of points of distance less than or equal to 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.
For example, a onedimensional sphere is the surface of what is commonly called a "circle", while such a circle's interior and surface together are the twodimensional ball. Similarly, a twodimensional sphere is the surface of the Euclidean solid known colloquially as a "sphere", while the interior and surface together are the threedimensional ball.
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.
Contents
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
General area and volume formulas
The classical equation of a unit sphere is that of the ellipsoid with a radius of 1 and no alterations to the x, y, or z axes:
 f(x,y,z) = x^{2} + y^{2} + z^{2} = 1
The volume of the unit ball in ndimensional Euclidean space, and the surface area of the unit sphere, appear in many important formulas of analysis. The volume of the unit ball in n dimensions, which we denote V_{n}, can be expressed by making use of the Gamma function. It is
where n!! is the double factorial.
The hypervolume of the (n–1)dimensional unit sphere (i.e., the "area" of the surface of the ndimensional unit ball), which we denote A_{n}, can be expressed as
where the last equality holds only for n > 0.
The surface areas and the volumes for some values of n are as follows:
n A_{n} (surface area) V_{n} (volume) 0 0(1 / 0!)π^{0} 0.000 (1 / 0!)π^{0} 1.000 1 1(2^{1} / 1!!)π^{0} 2.000 (2^{1} / 1!!)π^{0} 2.000 2 2(1 / 1!)π^{1} = 2π 6.283 (1 / 1!)π^{1} = π 3.142 3 3(2^{2} / 3!!)π^{1} = 4π 12.57 (2^{2} / 3!!)π^{1} = (4 / 3)π 4.189 4 4(1 / 2!)π^{2} = 2π^{2} 19.74 (1 / 2!)π^{2} = (1 / 2)π^{2} 4.935 5 5(2^{3} / 5!!)π^{2} = (8 / 3)π^{2} 26.32 (2^{3} / 5!!)π^{2} = (8 / 15)π^{2} 5.264 6 6(1 / 3!)π^{3} = π^{3} 31.01 (1 / 3!)π^{3} = (1 / 6)π^{3} 5.168 7 7(2^{4} / 7!!)π^{3} = (16 / 15)π^{3} 33.07 (2^{4} / 7!!)π^{3} = (16 / 105)π^{3} 4.725 8 8(1 / 4!)π^{4} = (1 / 3)π^{4} 32.47 (1 / 4!)π^{4} = (1 / 24)π^{4} 4.059 9 9(2^{5} / 9!!)π^{4} = (32 / 105)π^{4} 29.69 (2^{5} / 9!!)π^{4} = (32 / 945)π^{4} 3.299 10 10(1 / 5!)π^{5} = (1 / 12)π^{5} 25.50 (1 / 5!)π^{5} = (1 / 120)π^{5} 2.550 where the decimal expanded values for n ≥ 2 are rounded to the displayed precision.
Recursion
The A_{n} values satisfy the recursion:
 A_{0} = 0
 A_{1} = 2
 A_{2} = 2π
 for n > 2.
The V_{n} values satisfy the recursion:
 V_{0} = 1
 V_{1} = 2
 for n > 1.
Fractional dimensions
Main article: Hausdorff measureThe formulae for A_{n} and V_{n} can be computed for any real number n ≥ 0, and there are circumstances under which it is appropriate to seek the sphere area or ball volume when n is not a nonnegative integer.
Other radii
Main article: SphereThe surface area of an (n–1)dimensional sphere with radius r is A_{n} r^{n−1} and the volume of an ndimensional ball with radius r is V_{n} r^{n}. For instance, the area is A = 4π r^{ 2} for the surface of the threedimensional ball of radius r. The volume is V = 4π r^{ 3} / 3 for the threedimensional ball of radius r.
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,·),
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 R^{n}. 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. Here are some images of the unit ball for the twodimensional space for various values of p (the unit ball being concave for p < 1 and convex for p ≥ 1):
These illustrate why the condition p ≥ 1 is necessary in the definition of the norm, as the unit ball in any normed space must be convex as a consequence of the triangle inequality.
Note that for the circumferences C_{p} of the twodimensional unit balls we have:
 is the maximum value.
 is the minimum value.
Generalizations
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.
Quadratic forms
If V is a linear space with a real quadratic form F:V → R, then { p ∈ V : F(p) = 1 } is sometimes called the unit quasisphere of V. For example, the quadratic form x^{2} − y^{2} corresponds to a unit quasisphere that is a hyperbola which plays the role of the "unit circle" in the plane of splitcomplex numbers. Similarly, quadratic form x^{2} yields a pair of lines for the unit quasisphere in the dual number plane.
See also
 ball (mathematics)
 sphere
 Table of mathematical symbols
 unit circle
 unit disk
 unit sphere bundle
 unit square
References
 Mahlon M. Day (1958) Normed Linear Spaces, page 24, SpringerVerlag.
 Deza, E.; Deza, M. (2006), Dictionary of Distances, Elsevier, ISBN 0444520872. Reviewed in Newsletter of the European Mathematical Society 64 (June 2007), p. 57. This book is organized as a list of distances of many types, each with a brief description.
External links
 Weisstein, Eric W., "Unit sphere" from MathWorld.
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