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What is Half-space (geometry)?

In geometry, a half-space is either of the two parts into which a plane divides the three-dimensional Euclidean space. More generally, a half-space is either of the two parts into which a hyperplane divides an affine space. That is, the points that are not incident to the hyperplane are partitioned into two convex sets (i.e., half-spaces), such that any subspace connecting a point in one set to a point in the other must intersect the hyperplane. A half-space can be either open or closed. An open half-space is either of the two open sets produced by the subtraction of a hyperplane from the affine space. A closed half-space is the union of an open half-space and the hyperplane that defines it. If the space is two-dimensional, then a half-space is called a half-plane (open or closed). A half-space in a one-dimensional space is called a half-line or ray. A half-space may be specified by a linear inequality, derived from the linear equation that specifies the defining hyperplane. A strict linear inequality specifies an open half-space: A non-strict one specifies a closed half-space: Here, one assumes that not all of the real numbers a1, a2, ..., an are zero.

Technology Types

euclidean geometrygeometric shapegeometrypossessionproperties of topological spacespatial property


Affine halfspaceClosed half-spaceHalf planeHalf-planeHalfplaneLower half spaceLower half-spaceOpen half-spaceUpper half spaceUpper half-space


Demi-espace (fr)Espazioerdi (eu)Halbraum (de)Półprzestrzeń (pl)Semiespacio (es)Semiespai (ca)Semispazio (it)Півпростір (uk)Полупространство (ru)半空间 (zh)

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 — Date merged: 11/6/2021, 1:33:02 PM
 — Date scraped: 5/20/2021, 6:11:11 PM