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Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry, Elements. Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions ( theorems ) from these.
In modern geometry, Euclidean spaces are often defined from linear algebra. More precisely, a Euclidean space E is defined as a set to which is associated an inner product space of finite dimension over the reals E → , {\displaystyle {\overrightarrow {E}},} and a group action of the additive group of E → , {\displaystyle {\overrightarrow {E ...
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's Elements, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension n, which are called Euclidean n-spaces when one wants to specify their ...
In elliptic geometry we see a typical example of this. [1]: 108 In the spherical representation of elliptic geometry, lines are represented by great circles of a sphere with diametrically opposite points identified. In a different model of elliptic geometry, lines are represented by Euclidean planes passing through the origin. Even though these ...
The Euclidean distance gives Euclidean space the structure of a topological space, the Euclidean topology, with the open balls (subsets of points at less than a given distance from a given point) as its neighborhoods. [26] Comparison of Chebyshev, Euclidean and taxicab distances for the hypotenuse of a 3-4-5 triangle on a chessboard
For example, given a linear map T : V → W, the image T(V) of V, and the inverse image T −1 (0) of 0 (called kernel or null space), are linear subspaces of W and V, respectively. Another important way of forming a subspace is to consider linear combinations of a set S of vectors: the set of all sums
For example, a circle is a concept that makes sense in Euclidean geometry, but not in affine linear geometry or projective geometry, where circles cannot be distinguished from ellipses, since one may squeeze a circle to an ellipse. Similarly, a parabola is a concept in affine geometry but not in projective geometry, where a parabola is simply a ...
In Euclidean geometry, a plane is a flat two-dimensional surface that extends indefinitely. Euclidean planes often arise as subspaces of three-dimensional space. A prototypical example is one of a room's walls, infinitely extended and assumed infinitesimal thin.