Misplaced Pages

Half-space (geometry)

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
(Redirected from Halfplane) Bisection of Euclidean space by a hyperplane
This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.
Find sources: "Half-space" geometry – news · newspapers · books · scholar · JSTOR (December 2024) (Learn how and when to remove this message)

In geometry, a half-space is either of the two parts into which a plane divides the three-dimensional Euclidean space. 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.

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 1 x 1 + a 2 x 2 + ⋯ + a n x n > b {\displaystyle a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{n}x_{n}>b}

A non-strict one specifies a closed half-space:

a 1 x 1 + a 2 x 2 + ⋯ + a n x n ≥ b {\displaystyle a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{n}x_{n}\geq b}

Here, one assumes that not all of the real numbers a1, a2, ..., an are zero.

A half-space is a convex set.

See also

References

  1. "half-space". Merriam-Webster.com Dictionary. Merriam-Webster.
  2. ^ Weisstein, Eric W. "Half-Space". Wolfram MathWorld. Retrieved 2024-12-04.
  3. Weisstein, Eric W. "Half-Plane". Wolfram MathWorld. Retrieved 2024-12-04.
  4. "half line". Merriam-Webster.com Dictionary. Merriam-Webster.

External links

Category: