Orthant

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 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.

Read more about Orthant:  Standard Space