A hyperplane is a concept in geometry. It is a generalization of the concept of a plane. Table of Geometry, from the 1728 Cyclopaedia. ... In mathematics, a plane is the fundamental two-dimensional object. ...

In a one-dimensional space (such as a line), a hyperplane is a point; it divides a line into two rays. In two-dimensional space (such as the xy plane), a hyperplane is a line; it divides the plane into two half-planes. In three-dimensional space, a hyperplane is an ordinary plane; it divides the space into two half-spaces. This concept can also be applied to four-dimensional space and beyond, where the dividing object is simply referred to as a hyperplane. A spatial point is an entity with a location in space but no extent (volume, area or length). ... A line, or straight line, can be described as an (infinitely) thin, (infinitely) long, perfectly straight curve (the term curve in mathematics includes straight curves). In Euclidean geometry, exactly one line can be found that passes through any two points. ... A line, or straight line, can be described as an (infinitely) thin, (infinitely) long, perfectly straight curve (the term curve in mathematics includes straight curves). In Euclidean geometry, exactly one line can be found that passes through any two points. ... In geometry, a half-space is any of the two parts into which a hyperplane divides an affine space. ... Two intersecting planes in R3 In mathematics, a plane is a fundamental two-dimensional object. ... In geometry, a half-space is any of the two parts into which a hyperplane divides an affine space. ...

Formal definition

In the general case, a hyperplane is an affine subspace of codimension 1. In other words, a hyperplane is a higher-dimensional analog of a (two-dimensional) plane in three-dimensional space. In mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space. ... In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, and more generally to submanifolds in manifolds, and suitable subsets of algebraic varieties. ...

An affine hyperplane in n-dimensional space can be described by a non-degenerate linear equation of the following form: bnrgljabvkfjabvgkjavgbkjavgkjA linear equation is an equation involving only the sum of constants or products of constants and the first power of a variable. ...

a₁x₁ + a₂x₂ + ... + a_nx_n = b.

Here, non-degenerate means that not all the a_i are zero. If b=0, one obtains a linear hyperplane, which goes through the origin of the space.

The two half-spaces defined by a hyperplane in n-dimensional space are:

a₁x₁ + a₂x₂ + ... + a_nx_n ≤ b

and

a₁x₁ + a₂x₂ + ... + a_nx_n ≥ b.

Notes

The term realm has been advocated for a three-dimensional hyperplane in four-dimensional space, but this is not in common use.

See also

• hypersurface
• decision boundary
• ham sandwich theorem