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Concept
Unit sphere
Summary
In mathematics, a unit sphere is simply a sphere of radius one around a given center. More generally, it is the set of points of distance 1 from a fixed central point, where different norms can be used as general notions of "distance". A unit ball is the closed set of points of distance less than or equal to 1 from a fixed central point. Usually the center is at the origin of the space, so one speaks of "the unit ball" or "the unit sphere".
Special cases are the unit circle and the unit disk.
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.
In Euclidean space of n dimensions, the (n−1)-dimensional unit sphere is the set of all points which satisfy the equation
The n-dimensional open unit ball is the set of all points satisfying the inequality
and the n-dimensional closed unit ball is the set of all points satisfying the inequality
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:
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 volume of the unit ball in n dimensions, which we denote Vn, 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 boundary of the n-dimensional unit ball), which we denote An−1, can be expressed as
where the last equality holds only for n > 0. For example, is the "area" of the boundary of the unit ball , which simply counts the two points. Then is the "area" of the boundary of the unit disc, which is the circumference of the unit circle. is the area of the boundary of the unit ball , which is the surface area of the unit sphere .
The surface areas and the volumes for some values of are as follows:
where the decimal expanded values for n ≥ 2 are rounded to the displayed precision.
Official source
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