Howard's Law

2013-12-04T13:05:56Z

123.202.23.180:

{{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.

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]].

A flat is also called a ''linear [[manifold]]'' or ''linear [[variety (disambiguation)#Mathematics|variety]]''.

==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}}:
:<math>3x + 5y = 8.</math>
In three-dimensional space, a single linear equation involving {{mvar|x}}, {{mvar|y}}, and {{mvar|z}} defines a plane, while a pair of linear equations can be used to describe a line. In general, a linear equation in {{mvar|n}} variables describes a hyperplane, and a system of linear equations describes the [[intersection (set theory)|intersection]] of those hyperplanes. Assuming the equations are consistent and [[linearly independent]], a system of {{mvar|k}} equations describes a flat of dimension {{math|''n'' − ''k''}}.

===Parametric===
A flat can also be described by a system of linear [[parametric equation]]s. A line can be described by equations involving one [[parameter]]:

:<math>x=2+3t,\;\;\;\;y=-1+t\;\;\;\;z=\frac{3}{2}-4t</math>

while the description of a plane would require two parameters:

:<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 parameters {{math|''t''1, … , ''tk''}}.

==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>

If every line from the first flat is parallel to some line from the second flat, then these 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===
For two flats of dimensions {{math|''k''1}} and {{math|''k''2}} there exist the minimal flat which contains them, of dimension at most {{math|''k''1 + ''k''2 + 1}}. If two flats intersect, then the dimension of the containing flat equals to {{math|''k''1 + ''k''2 − }}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.

Though, the lattice of all flats is not a [[distributive lattice]].
If two lines {{math|ℓ1}} and {{math|ℓ2}} intersect, then {{math|ℓ1 ∩ ℓ2}} is a point. If {{mvar|p}} is a point not lying on the same plane, then {{math|1=(ℓ1 ∩ ℓ2) + ''p'' = (ℓ1 + ''p'') ∩ (ℓ2 + ''p'')}}, both representing a line. But when {{math|ℓ1}} and {{math|ℓ2}} 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==
Aforementioned facts do not depend on a structure of Euclidean space (namely, [[Euclidean distance]]) and are correct in any [[affine space]]. In a Euclidean space:
* There is the distance between a flat and a point.

* There is the distance between two flats, equal to 0 if they intersect.

* If two flats intersect, then there is the [[angle]] between two flats, which belongs to the {{closed-closed|0, π/2}} interval between 0 and the [[right angle]].

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==See also==
* [[N-dimensional space]]
* [[Matroid]]

==Notes==
{{Reflist}}

==References==
* [[Heinrich Guggenheimer]] (1977) ''Applicable Geometry'',page 7, Krieger, New York.
* {{citation
| last = Stolfi
| first = Jorge
| title = Oriented Projective Geometry
| publisher = [[Academic Press]]
| date = 1991
| isbn = 978-0-12-672025-9 }} 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].
* [http://planetmath.org/encyclopedia/LinearManifold.html PlanetMath: linear manifold]

== External links==
*{{MathWorld|urlname=Hyperplane|title=Hyperplane}}
*{{MathWorld|urlname=Flat|title=Flat}}

[[Category:Euclidean geometry]]
[[Category:Affine geometry]]
[[Category:Linear algebra]]

[[fr:hyperplan]]

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Howard's Law