Extended Euclidean algorithm: Difference between revisions

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→‎Proof: Removed definition of d, since it wasn't used elsewhere in the proof.
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{{wiktionary}}
In [[geometry]], as a [[plane (geometry)|plane]] has one less [[dimension (mathematics and physics)|dimension]] than [[space]], a '''hyperplane''' is a subspace of one dimension less than its [[ambient space]].
 
A hyperplane of an ''n''-dimensional space is a [[flat (geometry)|flat]] [[subset]] with dimension ''n''&nbsp;&minus;&nbsp;1. By its nature, it separates the space into two [[Half-space (geometry)|half space]]s.
 
== Technical description ==
In [[geometry]], a '''hyperplane''' of an [[n-dimensional space|''n''-dimensional space]] ''V'' is a "flat" subset of dimension ''n''&nbsp;&minus;&nbsp;1, or equivalently, of [[codimension]]&nbsp;1 in&nbsp;''V''; it may therefore be referred to as an (''n''&nbsp;&minus;&nbsp;1)-[[flat (geometry)|flat]] of&nbsp;''V''. The space ''V'' may be a [[Euclidean space]] or more generally an [[affine space]], or a [[vector space]] or a [[projective space]], and the notion of hyperplane varies correspondingly; in all cases however, any hyperplane can be given in [[coordinate]]s as the solution of a single (due to the "codimension&nbsp;1" constraint) [[algebraic equation]] of degree&nbsp;1 (due to the "flat" constraint). If ''V'' is a vector space, one distinguishes "vector hyperplanes" (which are subspaces, and therefore must pass through the origin) and "affine hyperplanes" (which need not pass through the origin; they can be obtained by [[translation (geometry)|translation]] of a vector hyperplane). A hyperplane in a Euclidean space separates that space into two [[Half-space (geometry)|half space]]s, and defines a [[reflection (mathematics)|reflection]] that fixes the hyperplane and interchanges those two half spaces.
 
== Dihedral angles ==
 
The [[dihedral angle]] between two non-parallel hyperplanes of a Euclidean space is the angle between the corresponding [[normal vector]]s. The product of the transformations in the two hyperplanes is a [[rotation (mathematics)|rotation]] whose axis is the [[Euclidean subspace|subspace]] of codimension&nbsp;2 obtained by intersecting the hyperplanes, and whose angle is twice the angle between the hyperplanes.
 
==Special types of hyperplanes==
Several specific types of hyperplanes are defined with properties that are well suited for particular purposes. Some of these specializations are described here.
 
=== Affine hyperplanes ===
 
An '''affine hyperplane''' is an [[affine space|affine subspace]] of [[codimension]] 1 in an [[affine space]].
In [[Cartesian coordinates]], such a hyperplane can be described with a single [[linear equation]] of the following form (where at least one of the <math>a_i</math>'s is non-zero):
 
:<math>a_1x_1 + a_2x_2  + \cdots + a_nx_n = b.\ </math>
 
In the case of a real affine space, in other words when the coordinates are real numbers, this affine space separates the space into two half-spaces, which are the [[connected component (topology)|connected component]]s of the [[complement (set theory)|complement]] of the hyperplane, and are given by the [[inequalities]]
 
:<math>a_1x_1 + a_2x_2  + \cdots + a_nx_n < b\ </math>
 
and
 
:<math>a_1x_1 + a_2x_2  + \cdots + a_nx_n > b.\ </math>
 
As an example, a point is a hyper plane in 1-dimensional space, a line is a hyperplane in 2-dimensional space, and a plane is a hyperplane in 3-dimensional space. A line in 3-dimensional space is not a hyperplane, and does not separate the space into two parts (the complement of such a line is connected).
 
Any hyperplane of a Euclidean space has exactly two unit normal vectors.
 
Affine hyperplanes are used to define decision boundaries in many [[machine learning]] algorithms such as linear-combination (oblique) [[Decision tree learning|decision trees]], and [[Perceptron]]s.
 
=== Vector hyperplanes ===
 
In a vector space, a vector hyperplane is a [[Linear subspace|subspace]] of codimension&nbsp;1, only possibly shifted from the origin by a vector. Such a hyperplane is the solution of a single [[linear equation]].
 
===Projective hyperplanes===
'''Projective hyperplanes''', are used in projective geometry.  Projective geometry can be viewed as affine geometry with [[vanishing point]]s (points at infinity) added. An affine hyperplane together with the associated points at infinity forms a projective hyperplane. One special case of a projective hyperplane is the '''infinite''' or '''ideal hyperplane''', which is defined with the set of all points at infinity.
 
In real projective space, a hyperplane does not divide the space into two parts; rather, it takes two hyperplanes to separate points and divide up the space.  The reason for this is that in real projective space, the space essentially "wraps around" so that both sides of a lone hyperplane are connected to each other.
 
==See also==
*[[hypersurface]]
*[[decision boundary]]
*[[ham sandwich theorem]]
*[[arrangement of hyperplanes]]
*[[separating hyperplane theorem]]
*[[supporting hyperplane theorem]]
 
==References==
{{Reflist}}
* [[Charles W. Curtis]] (1968) ''Linear Algebra'', page 62, [[Allyn & Bacon]], Boston.
* [[Heinrich Guggenheimer]] (1977) ''Applicable Geometry'', page 7, Krieger, Huntington ISBN 0-88275-368-1 .
* Victor V. Prasolov & VM Tikhomirov (1997,2001) ''Geometry'', page 22, volume 200 in ''Translations of Mathematical Monographs'', [[American Mathematical Society]], Providence ISBN 0-8218-2038-9 .
 
== External links ==
*{{MathWorld|urlname=Hyperplane|title=Hyperplane}}
*{{MathWorld|urlname=Flat|title=Flat}}
 
{{Dimension topics}}
 
[[Category:Euclidean geometry]]
[[Category:Affine geometry]]
[[Category:Linear algebra]]
[[Category:Projective geometry]]

Revision as of 18:03, 28 February 2014



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