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 $\mathbb {R} ^{n}$.

In numerical applications one may meet a real n-dimensional linear space V with a basis {v_i} such that the overlap matrix is not equal to the the identity matrix,

$g_{ij}\equiv \langle v_{i},v_{j}\rangle \neq \delta _{ij},\quad i,j=1,\ldots ,n$

where δ_ij is the Kronecker delta The inner product between two elements x and y of the space with component vectors {x_i} and {y_j} with respect to the basis {v_i} is

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

The overlap matrix g_{i j} is an example of a metric tensor. When the metric tensor is a constant, symmetric, positive definite, n×n matrix, the linear space V is in fact (isomorphic to) an n-dimensional Euclidean space. By a choice of a new basis for V the matrix g_{i j} can be transformed to the identity matrix; the new basis is an orthonormal basis. Hence a Euclidean space may be defined as a linear inner product space that contains a basis with the identity matrix as its overlap matrix. In non-linear (curved, non-Euclidean) spaces the metric tensor is a function of position and cannot be transformed to an identity matrix by a single transformation holding for the whole space.

The definition above of a Euclidean space does not completely agree with the space appearing in the geometry of Euclid. After all, it was almost 2000 years after Euclid wrote his Elements that Descartes introduced ordered 2-tuples to describe points in the plane. Descartes singled out a special point: the origin. In this point he erected two perpendicular axes (now known as Cartesian axes), the x and y axis. In Euclid's geometry there is no origin, all points are equal.

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, equivalent to a shift of origin. 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 translations. Also the orientation in space of an object is irrelevant for its geometric properties, so that Euclid, also implicitly, assumed rotational invariance as well. The set of affine motions forms a group, named the Euclidean group.

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. A general Euclidean space may be defined as an n-dimensional affine space with inner product. Although classical Euclidean geometry does not introduce explicitly an inner product, it does so implicitly by considering lengths of line segments and magnitudes of angles.

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 Euclidean, 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 that uses the common Mercator projection. However, such a chart gives wrong distances 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 embedded in higher-dimensional Euclidean spaces.