Half-space Hyperplane

转自:http://en.wikipedia.org/wiki/Half-space

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.

One can have open and closed half-spaces. An open half-space is either of the two open sets produced by the subtraction of a hyperplane from the affine space. Aclosed 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 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, while a non-strict one

specifies a closed half-space. Here, one assumes that not all of the real numbersa₁,a₂, ...,a_n are zero.

Properties

A half-space is a convex set.
Any convex set can be described as the (possibly infinite) intersection of half-spaces.

Upper and lower half-spaces

The open (closed) upper half-space is the half-space of all (x₁,x₂, ...,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).

转自:http://en.wikipedia.org/wiki/Half-space_(disambiguation)

half-space may refer to either of the following:

in geometry, thehalf-space (orhalf-plane)—part of an affine space
in typography, thehalf-space punctuation character

转自:http://en.wikipedia.org/wiki/Hyperplane

hyperplane is a concept in geometry. It is a generalization of the plane into a different number of dimensions.

A hyperplane of an n-dimensional space is a flat subset with dimension n − 1. By its nature, it separates the space into two half spaces.

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