Euclidean space

In mathematics, a Euclidean space is a vector space of dimension n over the field of real numbers, where n is a finite natural number not equal to zero. It is isomorphic to the space $\mathbb {R} ^{n}$ of ordered n-tuples (columns) of real numbers and hence is usually identified with the latter. In addition, a distance d(x,y) must be defined between any two elements x and y of a Euclidean space, i.e., a Euclidean space is a metric space.

The distance is defined by means of the following positive definite inner product,

$d(\mathbf {x} ,\mathbf {y} )\equiv \langle \mathbf {x} -\mathbf {y} ,\mathbf {x} -\mathbf {y} \rangle ^{\frac {1}{2}}\equiv \left[\sum _{i=1}^{n}(x_{i}-y_{i})^{2}\right]^{\frac {1}{2}},$

where x_i are the components of x and y_i of y. Thus, most commonly a Euclidean space is defined as the real inner product space $\mathbb {R} ^{n}$. A general inner product between x and y can be written as

$\langle \mathbf {x} ,\mathbf {y} \rangle =\sum _{ij=1}^{n}x_{i}g_{ij}y_{j},$

where g_{i j} is the metric tensor of the space. A Euclidean space has an inner product with

$g_{ij}=\delta _{ij},\,$

where δ is the Kronecker delta. The metric tensor of a n-dimensional Euclidean space is the n×n identity matrix.

The definition of Euclidean space does not completely agree with the space appearing in the geometry of Euclid. After all, it was almost 2000 years after Euclid's Elements when Descartes introduced ordered 2-tuples and 3-tuples to describe points in the plane and the 3-dimensional space.

One can introduce the following affine map on $\mathbb {R} ^{n}$:

$\mathbf {x} \mapsto \mathbf {x} '=\mathbf {A} \mathbf {x} +\mathbf {c} ,\quad \mathbf {x} ,\mathbf {x} '\in \mathbb {R} ^{n},$

where A is a real n×n matrix and c is an ordered n-tuple of real numbers. If A is an orthogonal matrix this map leaves distances invariant and is called an affine motion; if furthermore c = 0 it is a rotation. If A = E (the identity matrix), it is a translation. In the classical Euclidean geometry it is irrelevant at which points in space the geometrical objects (circles, triangles, Platonic solids, etc.) are located. This means that Euclid assumed implicitly the invariance of his geometry under the set of affine motions.

A real inner product space equipped with an affine map is an affine space. Thus, formally, the space of high-school geometry is the 2- or 3-dimensional affine space equipped with inner product and, in summary, a Euclidean space may be defined as an n-dimensional affine space with inner product.

Finally, it may be of interest to mention an example of a space that is not Euclidean, i.e., non-flat—the flatness being given by the definition of distance. The best known example of a curved space is the surface of the Earth. Locally the surface is flat, i.e., Euclidean, but globally it is curved. Somebody planning a day's hike will see the Earth as flat, but an airplane pilot planning a flight from Europe to the US will not. Most long-distance flights follow a great circle, because that is the shortest distance on the surface of a sphere. Planes do not fly along parallels of latitude (the equator excepted), even if the points of departure and destination are at the same latitude. Flying along a parallel seems shortest on a chart in an atlas. However, a chart gives the wrong distance because it approximates the curved surface of the Earth by a flat 2-dimensional Euclidean plane, see Riemannian manifold for more details about the distance on curved spaces.