thumb|In two dimensions, there are four orthants (called quadrants)
In geometry, an orthant or hyperoctant is the analogue in n-dimensional Euclidean space of a quadrant in the plane or an octant in three dimensions.
In general an orthant in n-dimensions can be considered the intersection of n mutually orthogonal half-spaces. 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> ≥ 0 ε<sub>2</sub>x<sub>2</sub> ≥ 0 · · · ε<sub>n</sub>x<sub>n</sub> ≥ 0,
where each ε<sub>i</sub> is +1 or −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> > 0 ε<sub>2</sub>x<sub>2</sub> > 0 · · · ε<sub>n</sub>x<sub>n</sub> > 0,
where each ε<sub>i</sub> is +1 or −1.
By dimension:
- In one dimension, an orthant is a ray.
- In two dimensions, an orthant is a quadrant.
- In three dimensions, an orthant is an octant.
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 facets, one per orthant.
The nonnegative orthant is the generalization of the first quadrant to n-dimensions and is important in many constrained optimization problems.
See also
- Cross polytope (or orthoplex) – a family of regular polytopes in n-dimensions which can be constructed with one simplex facets in each orthant space.
- Measure polytope (or hypercube) – a family of regular polytopes in n-dimensions which can be constructed with one vertex in each orthant space.
- Orthotope – generalization of a rectangle in n-dimensions, with one vertex in each orthant.
References
Further reading
- The facts on file: Geometry handbook, Catherine A. Gorini, 2003, , p.113
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