Definitions for euclidean space
This page provides all possible meanings and translations of the word euclidean space
Random House Webster's College Dictionary
ordinary two- or three-dimensional space.
Origin of Euclidean space:
a space in which Euclid's axioms and definitions apply; a metric space that is linear and finite-dimensional
Ordinary two- or three-dimensional space, characterised by an infinite extent along each dimension and a constant distance between any pair of parallel lines.
Any real vector space on which a real-valued inner product (and, consequently, a metric) is defined.
In mathematics, particularly in geometry, the concept of an Euclidean space encompasses Euclidean plane and the three-dimensional space of Euclidean geometry as spaces of dimensions 2 and 3 respectively. It is named after the Ancient Greek mathematician Euclid of Alexandria. The term “Euclidean” distinguishes these spaces from other types of spaces considered in modern geometry. Euclidean spaces also generalizes these ideas to higher dimensions. Classical Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain postulates, while the other properties of these spaces were deduced as theorems. Geometric constructions also used to define rational numbers. When algebra and mathematical analysis became developed enough, this relation reversed and now it is more common to define Euclidean space using Cartesian coordinates and the ideas of analytic geometry. It means that points of the space are specified with collections of real numbers and geometric shapes are defined as equations and inequalities. This approach brings the tools of algebra and calculus to bear on questions of geometry, and has the advantage that it generalizes easily to Euclidean spaces of more than three dimensions.
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