In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a half-space is either of the two parts into which a
plane divides the three-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
.
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
Ray may refer to:
Fish
* Ray (fish), any cartilaginous fish of the superorder Batoidea
* Ray (fish fin anatomy), a bony or horny spine on a fin
Science and mathematics
* Ray (geometry), half of a line proceeding from an initial point
* Ray (gr ...
''.
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 set
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex ...
s (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 set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
s 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.
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 ''a''
1, ''a''
2, ..., ''a''
''n'' are zero.
Properties
* A half-space is a
convex set
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex ...
.
Upper and lower half-spaces
The open (closed) upper half-space is the half-space of all (''x''
1, ''x''
2, ..., ''x''
''n'') such that ''x''
''n'' > 0 (≥ 0). The open (closed) lower half-space is defined similarly, by requiring that ''x''
''n'' be negative (non-positive).
See also
*
Line (geometry)
*
Upper half-plane
In mathematics, the upper half-plane, \,\mathcal\,, is the set of points in the Cartesian plane with > 0.
Complex plane
Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to ...
*
Poincaré half-plane model
*
Siegel upper half-space
*
Nef polygon, construction of
polyhedra using half-spaces.
External links
*
*
{{DEFAULTSORT:Half-Space
Euclidean geometry