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{{short description|Affine subspace of a Euclidean space}}
<!--{{redirect text|Euclidean subspace|It may erroneously refer to a [[linear subspace]] in the [[real n-space]]}} not redirected yet, but I hope it will come soon -->
{{redirect|Euclidean subspace|a subspace that contains the zero vector or a fixed origin|Linear subspace}}
{{refimprove|date=April 2013}}
In [[geometry]], a '''flat''' is a subset of [[n-dimensional space|{{mvar|n}}-dimensional space]] that is [[congruence (geometry)|congruent]] to a [[Euclidean space]] of lower [[dimension]]. The flats in two-dimensional space are [[point (mathematics)|points]] and [[line (mathematics)|lines]], and the flats in [[three-dimensional space]] are points, lines, and [[plane (mathematics)|planes]].
In {{mvar|n}}-dimensional space, there are flats of every dimension from 0 to {{math|''n'' − 1}}.<ref>In addition, all of {{mvar|n}}-dimensional space is sometimes considered an {{mvar|n}}-dimensional flat as a subset of itself.</ref> Flats of dimension {{math|''n'' − 1}} are called [[hyperplane]]s.
 
In [[geometry]], a '''flat''' or '''affine subspace''' is a subset of an [[affine space]] that is itself an affine space (of equal or lower [[dimension]]). In the case the parent space is [[Euclidean space|Euclidean]], a flat is a '''Euclidean subspace''' which inherits the notion of [[Euclidean distance|distance]] from its parent space.
Flats are similar to [[linear subspace]]s, except that they need not pass through the [[origin (mathematics)|origin]]. If Euclidean space is considered as an [[affine space]], the flats are precisely the [[affine subspace]]s. Flats are important in [[linear algebra]], where they provide a geometric realization of the solution set for a [[system of linear equations]].
 
The flats in a [[plane (geometry)|plane]] (two-dimensional space) are [[point (mathematics)|points]], [[line (mathematics)|lines]], and the plane itself; the flats in [[three-dimensional space]] are points, lines, planes, and the space itself. In an [[n-dimensional space|{{mvar|n}}-dimensional space]], there are {{mvar|k}}-flats of every dimension {{mvar|k}} from 0 to {{math|''n''}}; subspaces one dimension lower than the parent space, {{math|(''n''&thinsp;−&hairsp;1)}}-flats, are called ''[[hyperplane]]s''.
A flat is also called a ''linear [[manifold]]'' or ''linear [[variety (disambiguation)#Mathematics|variety]]''.
 
Flats occur in [[linear algebra]], as geometric realizations of solution sets of [[systems of linear equations]].
 
A flat is a [[manifold]] and an [[algebraic variety]], and is sometimes called a ''linear manifold'' or ''linear variety'' to distinguish it from other manifolds or varieties.
 
==Descriptions==
 
===By equations===
A flat can be described by a [[system of linear equations]]. For example, a line in two-dimensional space can be described by a single linear equation involving {{mvar|x}} and {{mvar|y}}:
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:<math>x=5+2t_1-3t_2,\;\;\;\; y=-4+t_1+2t_2\;\;\;\;z=5t_1-3t_2.\,\!</math>
 
In general, a parameterization of a flat of dimension {{mvar|k}} would require {{mvar|k}} parameters, e.g. {{math|''t''<sub>1</sub>, …  …, ''t<sub>k</sub>''}}.
 
==Operations and relations on flats==
 
===Intersecting, parallel, and skew flats===
An [[set intersection|intersection]] of flats is either a flat or the [[empty set]].<ref>Can be considered as {{num|−1}}-flat.</ref>
 
<!-- only for equal dimensions? -->If everyeach line from the firstone flat is parallel to some line from the secondanother flat, then these two flats are [[parallel (geometry)|parallel]]. Two parallel flats of the same dimension either coincide or do not intersect; they can be described by two systems of linear equations which differ only in their right-hand sides.
 
If flats do not intersect, and no line from the first flat is parallel to a line from the second flat, then these are [[skew flats]]. It is possible only if sum of their dimensions is less than dimension of the ambient space.
 
===Join<!-- is it a correct name? -->===
For two flats of dimensions {{math|''k''<sub>1</sub>}} and {{math|''k''<sub>2</sub>}} there existexists the minimal flat which contains them, of dimension at most {{math|''k''<sub>1</sub> + ''k''<sub>2</sub> + 1}}. If two flats intersect, then the dimension of the containing flat equals to {{math|''k''<sub>1</sub> + ''k''<sub>2</sub> − }} minus the dimension of the intersection.
 
===Properties of operations===
These two operations (referred to as ''meet'' and ''join'') make the set of all flats in the Euclidean {{mvar|n}}-space a [[lattice (order)|lattice]] and can build systematic coordinates for flats in any dimension, leading to [[Plücker coordinates|Grassmann coordinates]] or dual Grassmann coordinates. For example, a line in three-dimensional space is determined by two distinct points or by two distinct planes.
 
ThoughHowever, the lattice of all flats is not a [[distributive lattice]].
<!-- an image needed -->If two lines {{math|ℓ<sub>1</sub>}} and {{math|ℓ<sub>2</sub>}} intersect, then {{math|ℓ<sub>1</sub> ∩ ℓ<sub>2</sub>}} is a point. If {{mvar|p}} is a point not lying on the same plane, then {{math|1=(ℓ<sub>1</sub> ∩ ℓ<sub>2</sub>) + ''p'' = (ℓ<sub>1</sub> + ''p'') ∩ (ℓ<sub>2</sub> + ''p'')}}, both representing a line. But when {{math|ℓ<sub>1</sub>}} and {{math|ℓ<sub>2</sub>}} are parallel, this [[distributive property|distributivity]] fails, giving {{mvar|p}} on the left-hand side and a third parallel line on the right-hand side. The ambient space would be a [[projective space]] to accommodate intersections of parallel flats, which lead to [[point at infinity|objects "at infinity"]].
 
==Euclidean geometry==
AforementionedThe aforementioned facts do not depend on athe structure being that of Euclidean space (namely, involving [[Euclidean distance]]) and are correct in any [[affine space]]. In a Euclidean space:
* There is the distance between a flat and a point. (See for example ''[[Distance from a point to a plane]]'' and ''[[Distance from a point to a line]]''.)
<!-- write formulae -->
* There is the distance between two flats, equal to 0 if they intersect. (See for example ''[[Distance between two lines]]'' (in the same plane) and ''{{section link|Skew lines#Distance}}''.)
<!-- write formulae -->
* If two flats intersect, then thereThere is the [[angle]] between two flats, which belongs to the interval {{closed-closed|0, &pi;/2}} interval between 0 and the [[right angle]]. (See for example ''[[Dihedral angle]]'' (between two planes). See also ''[[Angles between flats]]''.)
<!-- the definition is tricky because it depends on dimension of the ambient space -->
<!-- once principal angles are defined ([[Angles between flats]]), the single angle is acos(prod(cos theta_i)). See http://dx.doi.org/10.1007/BF00148212 -->
{{expand section|date=April 2013}}
 
==See also==
* [[N-dimensional space]]
* [[Matroid]]
* [[Coplanarity]]
* [[Isometry]]
 
==Notes==
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==References==
* [[Heinrich Guggenheimer]] (1977), ''Applicable Geometry'',page 7, Krieger, New York, page 7.
* {{citation
| last = Stolfi
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| publisher = [[Academic Press]]
| date = 1991
| isbn = 978-0-12-672025-9 }}<br />From original [[Stanford]] Ph.D. dissertation, ''Primitives for Computational Geometry'', available as [http://www.hpl.hp.com/techreports/Compaq-DEC/SRC-RR-36.html DEC SRC Research Report 36] {{Webarchive|url=https://web.archive.org/web/20211017121735/https://www.hpl.hp.com/techreports/Compaq-DEC/SRC-RR-36.html |date=2021-10-17 }}.
* [http://planetmath.org/encyclopedia/LinearManifold.html PlanetMath: linear manifold]
 
== External links==
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[[Category:Linear algebra]]
 
[[fr:hyperplanHyperplan]]
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