Hyperplane

A hyperplane is not to be confused with a hypersonic aircraft.

A hyperplane is a concept in geometry. It is a generalization of the concept of a plane.

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.

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.

An affine hyperplane in n-dimensional space can be described by a non-degenerate linear equation of the following form:

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