nLab
Euclidean space

Euclidean spaces

Idea

A Euclidean space is a formulation in modern terms of the spaces that Euclid studied, equipped with the structures that Euclid recognised his spaces as having.

Definition

A Euclidean space is an affine inner product space. That is, it is an affine space EE modelled on a vector space VV which has an inner product.

One generally takes the inner product to be positive-definite; otherwise, we say that EE is only a pseudo-Euclidean space. Also, one generally takes the dimension to be finite; Euclid himself only considered dimensions up to 33. For an infinite-dimensional Euclidean space, you would probably want VV to be a Hilbert space.

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.

Revised on August 31, 2011 17:28:37 by Toby Bartels (75.88.81.61)