Half-space

In geometry Geometry

Geometry arose as the field of knowledge dealing with spatial relationships.... 

, a half-space is any of the two parts into which a plane divides the three-dimensional space. More generally, a half-space is any of the two parts into which a hyperplane divides an affine space. One can have open and closed half-spaces. An open half-space is any of the two open set Open set

In topology [i] and related fields of mathematics [i], a set [i] U is called open if, intuitively sp ... 

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 .

Discussions

Discussion Features

Ask a question about 'Half-space'    Start a new discussion about 'Half-space'    Answer questions about 'Half-space'    'Half-space' discussion forum

Encyclopedia

In geometry Geometry

Geometry arose as the field of knowledge dealing with spatial relationships.... 

, a half-space is any of the two parts into which a plane divides the three-dimensional space. More generally, a half-space is any of the two parts into which a hyperplane divides an affine space.

One can have open and closed half-spaces. An open half-space is any of the two open set Open set

In topology [i] and related fields of mathematics [i], a set [i] U is called open if, intuitively sp ... 

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 . A half-space in a one-dimensional Dimension

In common usage, a dimension is a parameter [i] or measurement [i] required to define the characteristi ... 

 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 Inequality

In mathematics [i], an inequality is a statement about the relative size or order of two objects.
... 

a₁x₁ + a₂x₂ + ... + a_nx_n > b

specifies an open half-space, while a non-strict one

a₁x₁ + a₂x₂ + ... + a_nx_n b

specifies a closed half-space. Here, one assumes that not all of the real numbers a₁, a₂, ..., a_n are zero.

Properties

• A half-space is a convex set Convex set
  
  In Euclidean space [i], an object is convex if for every pair of points within the object, every point o ... 
  
  .
  

Upper and lower half-spaces

The open upper half-space is the half-space of all such that x_n >0 . The open lower half-space is defined similarly, by requiring that x_n be negative .

See also

• upper half-plane
• Poincaré half-plane model
  

Categories:

• Euclidean geometry