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Generalization of a quadrant to any dimension From Wikipedia, the free encyclopedia
In geometry, an orthant[1] or hyperoctant[2] 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 2n orthants in n-dimensional space.
More specifically, a closed orthant in Rn is a subset defined by constraining each Cartesian coordinate to be nonnegative or nonpositive. Such a subset is defined by a system of inequalities:
where each εi is +1 or −1.
Similarly, an open orthant in Rn is a subset defined by a system of strict inequalities
where each εi is +1 or −1.
By dimension:
John Conway and Neil Sloane defined the term n-orthoplex from orthant complex as a regular polytope in n-dimensions with 2n simplex facets, one per orthant.[3]
The nonnegative orthant is the generalization of the first quadrant to n-dimensions and is important in many constrained optimization problems.
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