Orthant

From Wikipedia, the free encyclopedia

In geometry, a closed orthant is one of the 2ⁿ subsets of an n-dimensional Euclidean space defined by constraining each Cartesian coordinate axis to be nonnegative or nonpositive. That is, a closed orthant is the analogue of a closed quadrant in the plane and a closed octant in three-dimensional space. A closed orthant is defined by a system of inequalities

ε_ix_i ≥ 0 for 1 ≤ i ≤ n

on the coordinates x_i, where each ε_i is +1 or −1.

An open orthant is similar, except coordinates are constrained to be positive or negative (with defining inequalities ε_ix_i > 0 for 1 ≤ i ≤ n).