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 one-dimensional sphere is the surface of what is commonly called a "circle", while such a circle's interior and surface together are the two-dimensional ball. Similarly, a two-dimensional sphere is the surface of the Euclidean solid known colloquially as a "sphere", while the interior and surface together are the three-dimensional 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.

In Euclidean space of n dimensions, the unit sphere is the set of all points ${\displaystyle (x_{1},\ldots ,x_{n})}$ which satisfy the equation

The open unit ball is the set of all points satisfying the inequality

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