Editing Euclidean space (section)

==Affine structure==
{{Main|Affine space}}

Some basic properties of Euclidean spaces depend only on the fact that a Euclidean space is an [[affine space]]. They are called [[affine property|affine properties]] and include the concepts of lines, subspaces, and parallelism, which are detailed in next subsections.

===Subspaces===
{{main|Flat (geometry)}}
Let {{mvar|E}} be a Euclidean space and <math>\overrightarrow E</math> its associated vector space.

A ''flat'', ''Euclidean subspace'' or ''affine subspace'' of {{mvar|E}} is a subset {{mvar|F}} of {{mvar|E}} such that

<math display="block">\overrightarrow F = \Bigl\{\overrightarrow {PQ} \mathrel{\Big|} P\in F, Q\in F \Bigr\}\vphantom{\frac({}}</math>

as the associated vector space of {{mvar|F}} is a [[linear subspace]] (vector subspace) of <math>\overrightarrow E.</math> A Euclidean subspace {{mvar|F}} is a Euclidean space with <math>\overrightarrow F</math> as the associated vector space. This linear subspace <math>\overrightarrow F</math> is also called the ''direction'' of {{mvar|F}}.

If {{mvar|P}} is a point of {{mvar|F}} then

<math display="block">F = \Bigl\{P+v \mathrel{\Big|} v\in \overrightarrow F \Bigr\}.</math>

Conversely, if {{mvar|P}} is a point of {{mvar|E}} and <math>\overrightarrow V</math> is a [[linear subspace]] of <math>\overrightarrow E,</math> then

<math display="block">P + \overrightarrow V = \Bigl\{P + v \mathrel{\Big|} v\in \overrightarrow V \Bigr\}</math>

is a Euclidean subspace of direction <math>\overrightarrow V</math>. (The associated vector space of this subspace is <math>\overrightarrow V</math>.)

A Euclidean vector space <math>\overrightarrow E</math> (that is, a Euclidean space that is equal to <math>\overrightarrow E</math>) has two sorts of subspaces: its Euclidean subspaces and its linear subspaces. Linear subspaces are Euclidean subspaces and a Euclidean subspace is a linear subspace if and only if it contains the zero vector.

===Lines and segments===
{{main|Line (geometry)}}
In a Euclidean space, a ''line'' is a Euclidean subspace of dimension one. Since a vector space of dimension one is spanned by any nonzero vector, a line is a set of the form

<math display="block">\Bigl\{P + \lambda \overrightarrow{PQ} \mathrel{\Big|} \lambda \in \R \Bigr\},\vphantom{\frac({}}</math>

where {{mvar|P}} and {{mvar|Q}} are two distinct points of the Euclidean space as a part of the line.

It follows that ''there is exactly one line that passes through (contains) two distinct points.'' This implies that two distinct lines intersect in at most one point.

A more symmetric representation of the line passing through {{mvar|P}} and {{mvar|Q}} is

<math display="block">\Bigl\{O + (1-\lambda)\overrightarrow{OP} + \lambda \overrightarrow{OQ} \mathrel{\Big|} \lambda \in \R \Bigr\},\vphantom{\frac({}}</math>

where {{mvar|O}} is an arbitrary point (not necessary on the line).

In a Euclidean vector space, the zero vector is usually chosen for {{mvar|O}}; this allows simplifying the preceding formula into

<math display="block">\bigl\{(1-\lambda) P + \lambda Q \mathrel{\big|} \lambda \in \R\bigr\}.</math>

A standard convention allows using this formula in every Euclidean space, see {{slink|Affine space|Affine combinations and barycenter}}.

The ''[[line segment]]'', or simply ''segment'', joining the points {{mvar|P}} and {{mvar|Q}} is the subset of points such that {{math|0 ≤ ''{{lambda}}'' ≤ 1}} in the preceding formulas. It is denoted {{mvar|PQ}} or {{mvar|QP}}; that is

<math display="block">PQ = QP = \Bigl\{P+\lambda \overrightarrow{PQ} \mathrel{\Big|} 0 \le \lambda \le 1\Bigr\}.\vphantom{\frac({}}</math>

===Parallelism===
{{main|Parallel (geometry)}}

Two subspaces {{mvar|S}} and {{mvar|T}} of the same dimension in a Euclidean space are ''parallel'' if they have the same direction (i.e., the same associated vector space).{{efn|It may depend on the context or the author whether a subspace is parallel to itself}} Equivalently, they are parallel, if there is a translation vector {{mvar|v}} that maps one to the other:

<math display="block">T= S+v.</math>

Given a point {{mvar|P}} and a subspace {{mvar|S}}, there exists exactly one subspace that contains {{mvar|P}} and is parallel to {{mvar|S}}, which is <math>P + \overrightarrow S.</math> In the case where {{mvar|S}} is a line (subspace of dimension one), this property is [[Playfair's axiom]].

It follows that in a Euclidean plane, two lines either meet in one point or are parallel.

The concept of parallel subspaces has been extended to subspaces of different dimensions: two subspaces are parallel if the direction of one of them is contained in the direction to the other.

{{anchor|Euclidean norm}}