Editing Orthant

{{Short description|Generalization of a quadrant to any dimension}}
[[File:Cartesian coordinates 2D.svg|thumb|In two dimensions, there are four orthants (called quadrants)]]

In [[geometry]], an '''orthant'''<ref>{{cite book |first=Steven |last=Roman |authorlink=Steven Roman |title=Advanced Linear Algebra |location=New York |publisher=Springer |edition=2nd |year=2005 |isbn=0-387-24766-1 |url=https://books.google.com/books?id=FV_s8W58D4UC&pg=PA394 }}</ref> or '''hyperoctant'''<ref>{{MathWorld|title=Hyperoctant|urlname=Hyperoctant}}</ref> is the analogue in ''n''-dimensional [[Euclidean space]] of a [[Quadrant (plane geometry)|quadrant]] in the plane or an [[octant (solid geometry)|octant]] in three dimensions.

In general an orthant in ''n''-dimensions can be considered the intersection of ''n'' mutually orthogonal [[Half-space (geometry)|half-space]]s. By independent selections of half-space signs, there are 2<sup>''n''</sup> orthants in ''n''-dimensional space.

More specifically, a '''closed orthant''' in '''R'''<sup>''n''</sup> is a subset defined by constraining each [[Cartesian coordinate]] to be nonnegative or nonpositive.  Such a subset is defined by a system of inequalities:
:ε<sub>1</sub>''x''<sub>1</sub>&nbsp;≥&nbsp;0 &nbsp;&nbsp;&nbsp;&nbsp; ε<sub>2</sub>''x''<sub>2</sub>&nbsp;≥&nbsp;0 &nbsp;&nbsp;&nbsp;&nbsp;·&nbsp;·&nbsp;·&nbsp;&nbsp;&nbsp;&nbsp; ε<sub>''n''</sub>''x''<sub>''n''</sub>&nbsp;≥&nbsp;0,
where each ε<sub>''i''</sub> is +1 or &minus;1.

Similarly, an '''open orthant''' in '''R'''<sup>''n''</sup> is a subset defined by a system of strict inequalities
:ε<sub>1</sub>''x''<sub>1</sub>&nbsp;>&nbsp;0 &nbsp;&nbsp;&nbsp;&nbsp; ε<sub>2</sub>''x''<sub>2</sub>&nbsp;>&nbsp;0 &nbsp;&nbsp;&nbsp;&nbsp;·&nbsp;·&nbsp;·&nbsp;&nbsp;&nbsp;&nbsp; ε<sub>''n''</sub>''x''<sub>''n''</sub>&nbsp;>&nbsp;0,
where each ε<sub>''i''</sub> is +1 or −1.

By dimension:
*In one dimension, an orthant is a [[Line (mathematics)#Ray|ray]].
*In two dimensions, an orthant is a [[Cartesian coordinate system#Quadrants and octants|quadrant]].
*In three dimensions, an orthant is an [[octant (solid geometry)|octant]].

[[John Horton Conway|John Conway]] and [[Neil Sloane]] defined the term ''n''-[[orthoplex]] from '''orthant complex''' as a [[regular polytope]] in ''n''-dimensions with 2<sup>''n''</sup> [[simplex]] [[Facet (geometry)|facet]]s, one per orthant.<ref>{{cite book |first1=J. H. |last1=Conway |first2=N. J. A. |last2=Sloane |chapter=The Cell Structures of Certain Lattices |title=Miscellanea Mathematica |editor-last=Hilton |editor-first=P. |editor2-last=Hirzebruch |editor2-first=F. |editor3-last=Remmert |editor3-first=R. |publisher=Springer |location=Berlin |pages=89–90 |year=1991 |doi=10.1007/978-3-642-76709-8_5 |isbn=978-3-642-76711-1 }}</ref>

The '''nonnegative orthant''' is the generalization of the first [[Quadrant (plane geometry)|quadrant]] to ''n''-dimensions and is important in many [[constrained optimization]] problems.

==See also==
* [[Cross polytope]] (or orthoplex) – a family of [[regular polytope]]s in ''n''-dimensions which can be constructed with one [[simplex]] [[Facet (geometry)|facets]] in each orthant space.
* [[Measure polytope]] (or hypercube) – a family of regular polytopes in ''n''-dimensions which can be constructed with one [[vertex (geometry)|vertex]] in each orthant space.
* [[Orthotope]] – generalization of a rectangle in ''n''-dimensions, with one vertex in each orthant.

==References==
{{reflist}}

==Further reading==
* ''The facts on file: Geometry handbook'', Catherine A. Gorini, 2003, {{isbn|0-8160-4875-4}}, p.113

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

[[zh:卦限]]