Half-space (geometry)
{{Short description|Bisection of Euclidean space by a hyperplane}}{{More citations needed|date=December 2024}}
In geometry, a half-space is either of the two parts into which a plane divides the three-dimensional Euclidean space.{{Cite Merriam-Webster|half-space}} If the space is two-dimensional, then a half-space is called a half-plane (open or closed).{{Cite web |last=Weisstein |first=Eric W. |title=Half-Space |url=https://mathworld.wolfram.com/Half-Space.html |access-date=2024-12-04 |website=Wolfram MathWorld |language=en}}{{Cite web |last=Weisstein |first=Eric W. |title=Half-Plane |url=https://mathworld.wolfram.com/Half-Plane.html |access-date=2024-12-04 |website=Wolfram MathWorld |language=en}} A half-space in a one-dimensional space is called a half-line{{Cite Merriam-Webster|half line}} or ray.
More generally, a half-space is either of the two parts into which a hyperplane divides an n-dimensional 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.
The open (closed) upper half-space is the half-space of all (x1, x2, ..., xn) such that xn > 0 (≥ 0). The open (closed) lower half-space is defined similarly, by requiring that xn be negative (non-positive).
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.
A half-space is a convex set.
See also
- Hemisphere (geometry)
- Line (geometry)
- Poincaré half-plane model
- Siegel upper half-space
- Nef polygon, construction of polyhedra using half-spaces.
References
{{Reflist}}
External links
- {{springer|title=Half-plane|id=p/h046170}}
- {{Mathworld | urlname=Half-Space | title=Half-Space }}
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