Bisection method: Difference between revisions
en>Glrx →External links: article already has psuedo code |
en>Glenn L m Reverted 2 edits by 180.233.122.180 (talk) to last revision by Jitse Niesen. (TW) |
||
| Line 1: | Line 1: | ||
In [[geometry]], a '''half-space''' is either of the two parts into which a [[plane (geometry)|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 [[partition (set theory)|partitioned]] into two [[convex set]]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]]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. | |||
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 '''[[Line_(mathematics)#Ray|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 (mathematics)|inequality]] specifies an open half-space: | |||
:<math>a_1x_1+a_2x_2+\cdots+a_nx_n>b</math> | |||
A non-strict one specifies a closed half-space: | |||
:<math>a_1x_1+a_2x_2+\cdots+a_nx_n\geq b</math> | |||
Here, one assumes that not all of the real numbers ''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''n''</sub> 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 (''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>) such that ''x''<sub>''n''</sub> > 0 (≥ 0). The open (closed) '''lower half-space''' is defined similarly, by requiring that ''x''<sub>''n''</sub> be negative (non-positive). | |||
==See also== | |||
* [[Half-line]] | |||
* [[Upper half-plane]] | |||
* [[Poincaré half-plane model]] | |||
* [[Siegel upper half-space]] | |||
* [[Nef polygon]] , construction of [[polyhedra]] using half-spaces. | |||
==External links== | |||
* {{Mathworld | urlname=Half-Space | title=Half-Space }} | |||
{{DEFAULTSORT:Half-Space}} | |||
[[Category:Euclidean geometry]] | |||
Revision as of 11:21, 31 January 2014
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.
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.
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:
A non-strict one specifies a closed half-space:
Here, one assumes that not all of the real numbers a1, 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).
See also
- Half-line
- Upper half-plane
- Poincaré half-plane model
- Siegel upper half-space
- Nef polygon , construction of polyhedra using half-spaces.
External links
- 22 year-old Systems Analyst Rave from Merrickville-Wolford, has lots of hobbies and interests including quick cars, property developers in singapore and baking. Always loves visiting spots like Historic Monuments Zone of Querétaro.
Here is my web site - cottagehillchurch.com