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Unit ball : ウィキペディア英語版
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.
==Unit spheres and balls in Euclidean space==

In Euclidean space of ''n'' dimensions, the unit sphere is the set of all points (x_1, \ldots, x_n) which satisfy the equation
: x_1^2 + x_2^2 + \cdots + x_n ^2 = 1.
The open unit ball is the set of all points satisfying the inequality
: x_1^2 + x_2^2 + \cdots + x_n ^2 < 1,
and the closed unit ball is the set of all points satisfying the inequality
: x_1^2 + x_2^2 + \cdots + x_n ^2 \le 1.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
ウィキペディアで「Unit sphere」の詳細全文を読む



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