Hyperplane

In geometry, a hyperplane is a generalisation of a normal two-dimensional plane in three-dimensional space to its (n − 1)-dimensional analogue in n-dimensional space, where n is an arbitrary number. Specifically, it is an affine subspace of codimension 1. It can be described by a linear equation of the following form:-

a₁x₁ + a₂x₂ + ... + a_nx_n = b

This equation reduces the number of degrees of freedom of the point (x₁, x₂, ... , x_n) by 1, so it describes an (n − 1)-dimensional hyperplane. Of course, the number of degrees of freedom can be further restricted to produce a hyperplane of a lower number of dimensions (except in the base case where n = 1), but when discussing n-dimensional space the unmodified term "hyperplane" usually denotes an (n − 1)-dimensional hyperplane.

A zero-dimensional hyperplane is a point; a one-dimensional hyperplane is a (straight) line; and a two-dimensional hyperplane is a plane. The term realm has been advocated for a three-dimensional hyperplane, but this is not in common use.

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