Euclidean space

A Euclidean space or, more precisely, a Euclidean n-space is the generalization of the notions "plane" and "space" (from elementary geometry) to arbitrary dimensions n. Thus Euclidean 2-space is the plane, and Euclidean 3-space is space.

This generalization is obtained by extending the axioms of Euclidean geometry to allow n directions which are mutual perpendicular to each other. For practical purposes, Cartesian coordinates are introduced just as for 2 or 3 dimensions: Because of the larger dimension, n coordinates are needed to identify a point of the space. This approach is called "analytic geometry" because it allows to use the methods of linear algebra to solve geometrical questions by calculation with real numbers.

Euclidean space

Two- and three-dimensional geometry, as it is taught in school all over the world, was first described by Euclid more than two thousand years ago in his The Elements and is still useful for dealing with physical space even though modern physics has shown that geometry in the universe is more complicated.

This so-called Euclidean space is based on a few fundamental concepts, the notions point, straight line, plane and how they are related.
Two points determine a straight line (and a line segment), and a line and a point determine a line through that point and parallel to the given line. a line and a point (not on that line) determine a plane, and a plane and a point (not on that plane) "generate" 3-space.
Moreover, line segments have length, and the angle between intersecting lines can also be measured.

While it is difficult to picture higher dimensional objects it is easy to extend the mathematical concepts beyond three dimensions. In 4-dimensional space a plane and a point not on the plane determine a subspace, while a subspace and a point outside it generates 4-space. This can be iterated until a (n-1)-dimensional subspace (called hyperplane) and an exterior point are sufficient to generate the whole (n-dimensional) space.

Cartesian coordinates

A Euclidean space $\mathbb {E} ^{n}$ is a space of dimension n, where n is a finite natural number not equal to zero.

The n-dimensional Euclidean space is in one-to-one correspondence to the vector space ℝⁿ consisting of ordered n-tuples (columns) of real numbers. The definition of a 1-1 map between the two spaces is by choosing a point of $\mathbb {E} ^{n}$, the origin and erecting a set of axes in that point. Any point of $\mathbb {E} ^{n}$ obtains a unique set of coordinates with respect to these axes and accordingly is represented by an ordered set of real numbers, i.e., by an element of ℝⁿ. Conversely, given a column of n real numbers and a set of axes crossing in an origin, an element of $\mathbb {E} ^{n}$ (a "point") is determined uniquely. In fact, the two spaces are so closely related that they are often identified; in that case ℝⁿ is usually referred to as Euclidean space. However, strictly speaking ℝⁿ is not exactly the space appearing in Euclid's geometry, not even for n = 2 or n = 3. After all, it was almost 2000 years after Euclid wrote his Elements that Descartes introduced in 1637 ordered 2- and 3-tuples, now known as Cartesian coordinates, to describe points in the plane and in space. In Euclid's geometry there is no origin, all points are equal.

The definition of Euclidean space further requires a distance d(x,y) between any two of its elements x and y, i.e., a Euclidean space is an example of a metric space. The distance is defined by means of the following positive definite inner product on ℝⁿ,

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

where x_i are the components of x and y_i of y. Further, ⟨a, b⟩ stands for an inner product between a and b. Thus, a common definition of Euclidean space is that it is the linear space ℝⁿ equipped with positive definite inner product.

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 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 global transformation, i.e., by a single transformation holding on the whole space.

One can introduce the following affine map on ℝⁿ:

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

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. 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.