nLab
Euclidean space

Context

Analysis

Geometry

Euclidean spaces

Idea

The concept of Euclidean space in analysis, topology and differential geometry and specifically Euclidean geometry is a fomalization in modern terms of the spaces studied in Euclid 300BBC, equipped with the structures that Euclid recognised his spaces as having.

In the strict sense of the word, Euclidean space E nE^n of dimension nn is, up to isometry, the metric space whose underlying set is the Cartesian space n\mathbb{R}^n and whose distance function dd is given by the Euclidean norm:

d Eucl(x,y)xy= i=1 n(y ix i) 2. d_{Eucl}(x,y) \coloneqq {\Vert x-y\Vert} = \sqrt{ \sum_{i = 1}^n (y_i - x_i)^2 } \,.

(In Euclid 300BC this is considered for n=3n = 3.)

This means that in a Euclidean space one may construct for instance the unit sphere around any point, or the shortest curve connecting any two points. These are the operations studied in (Euclid 300BC), see at Euclidean geometry.

Of course these operations may be considered in every (other) metric space, too, see at non-Euclidean geometry. Euclidean geometry is distinguished notably from elliptic geometry or hyperbolic geometry by the fact that it satisfies the parallel postulate.

In regarding E n=( n,d Eucl)E^n = (\mathbb{R}^n, d_{Eucl}) (only) as a metric space, some extra structure still carried by n\mathbb{R}^n is disrgarded, such as its vector space structure, hence its affine structure? and its canonical inner product space structure. Sometimes “Euclidean space” is used to refer to E nE^n with that further extra structure remembered.

Retaining the inner product on top of the metric space structure means that on top of distances one may also speak of angles in a Euclidean space.

Then of course n\mathbb{R}^n carries also non-canonical inner product space strctures, not corresponding to the Euclidean norm. Regarding E nE^n as equipped with these one says that it is a pseudo-Euclidean space. These are now, again in the sense of Cartan geometry, the local model spaces for pseudo-Riemannian geometry.

Finally one could generalize and allow the dimension to be countably infinite, and regard seperable Hilbert spaces as generalized Eclidean spaces.

Remarks on terminology

Arguably, the spaces studied by Euclid were not really modelled on inner product spaces, as the distances were lengths, not real numbers (which, if non-negative, are ratios of lengths). So we should say that VV has an inner product valued in some oriented line LL (or rather, in L 2L^2). Of course, Euclid did not use the inner product (which takes negative values) directly, but today we can recover it from what Euclid did discuss: lengths (valued in LL) and angles (dimensionless).

Since the days of René Descartes, it is common to identify a Euclidean space with a Cartesian space, that is n\mathbb{R}^n for nn the dimension. But Euclid's spaces had no coordinates; and in any case, what we do with them is still coordinate-independent.

Lengths and angles

Given two points xx and yy of a Euclidean space EE, their difference xyx - y belongs to the vector space VV, where it has a norm

xy=xy,xy. {\|x - y\|} = \sqrt{\langle{x - y, x - y}\rangle} .

This real number (or properly, element of the line LL) is the distance between xx and yy, or the length of the line segment xy¯\overline{x y}. This distance function makes EE into an (LL-valued) metric space.

Given three points x,y,zx, y, z, with x,yzx, y \ne z (so that xz,yz0{\|x - z\|}, {\|y - z\|} \ne 0), we can form the ratio

xz,yzxzyz, \frac{\langle{x - z, y - z}\rangle}{{\|x - z\|} {\|y - z\|}} ,

which is a (dimensionless) real number. By the Cauchy–Schwartz inequality, this number lies between 1-1 and 11, so it's the cosine? of a unique angle measure between 00 and π\pi radians. This is the measure of the angle xzy\angle x z y. In a 22-dimensional Euclidean space, we can interpret xzy\angle x z y as a signed angle (so taking values anywhere on the unit circle) if we fix an orientation of EE.

Conversely, knowing angles and lengths, we may recover the inner product on VV;

xz,yz=xz¯yz¯cosxzy, \langle{x - z, y - z}\rangle = {\|\overline{x z}\|} {\|\overline{y z}\|} \cos \angle x z y ,

and other inner products are recovered by linearity. (We must then use the axioms of Euclidean geometry to prove that this is well defined and actually an inner product.) It’s actually possible to recover the inner product and angles from lengths alone; this is discussed at Hilbert space.

References

Revised on May 12, 2017 05:55:28 by Urs Schreiber (92.218.150.85)