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, ...,an 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, ...,xn) such that xn > 0 (≥ 0). The open (closed)lower half-space is defined similarly, by requiring that xn 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.