Euclidean Space

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euclidean space

[yü′klid·ē·ən ′spās]
A space consisting of all ordered sets (x1, …, xn ) of n numbers with the distance between (x1, …, xn ) and (y1, …, yn ) being given by the number n is called the dimension of the space.

Euclidean Space


in mathematics, a space whose properties are described by the axioms of Euclidean geometry. In a more general sense, a Euclidean space is an n-dimensional vector space, into which several special Cartesian coordinates can be introduced so that its metric is defined in the following manner: If point M has the coordinates (x1x2, …, xn and point M* has the coordinates (x1*, x2*, …, xn*), then the distance between these points is

References in periodicals archive ?
0001) in Euclidean spaces will be recognized as 1 in Hamming spaces.
In Section 3 the results of Section 2 are used to obtain some monocoronal tilings in higher dimensional Euclidean space with small translation groups.
Let M be an n-dimensional (n [greater than or equal to] 3) submanifold of a Euclidean space [E.
infinity]] (as we shall see these are, respectively, the "perfect substitutes" and "perfect complements" cases) and to the familiar Euclidean space [L.
The union of the two orthogonal proper (or classical) Euclidean 3-spaces yields a compound six-dimensional proper (or classical) Euclidean space with mutually orthogonal dimensions [x.
1 on a differential manifold in such a way that the properties of self-concordant functions in Euclidean space are preserved.
f[element of]F)] is a family of Euclidean spaces with dimensions n or less; for n = 3, [{X(f)}.
In case of Euclidean spaces, this heuristic is verifiable for some distributions simply because the farthest point from any given point is most likely to be a point that is close to the corner (or sides of the Euclidean hypercube).
The term "manifold" covers somewhat more complicated types than Euclidean spaces.
The following lemma is a corollary of Blumenthal's solution of the problem of isometric embedding of semimetric spaces in the Euclidean spaces, see [[6], p.
of California-Los Angeles) presents a textbook based on his introductory one-quarter graduate course on real analysis, focusing particularly on the basics of measure and integration theory both in Euclidean spaces and in abstract measure spaces.
G] of Euclidean spaces [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] underlying a connected graph G defined by