Orthant

Orthant
In two dimensions, there are 4 orthants (called quadrants)

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 permutations 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:

ε1x1 ≥ 0      ε2x2 ≥ 0     · · ·     εnxn ≥ 0,

where each εi is +1 or −1.

Similarly, an open orthant in Rn is a subset defined by a system of strict inequalities

ε1x1 > 0      ε2x2 > 0     · · ·     εnxn > 0,

where each εi is +1 or −1.

By dimension:

  1. In one dimension, an orthant is a ray.
  2. In two dimensions, an orthant is a quadrant.
  3. In three dimensions, an orthant is an octant.

Standard space

The n-orthant is a standard space in two ways: every polytope with n faces maps into the n-orthant via slack variables, and conversely every polygonal cone on n vertices is the image of (maps from) the n-orthant. Compare the n-simplex, which maps to every polytope with n-vertices.

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.

Notes


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