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:-

*a*_{1}*x*_{1}+*a*_{2}*x*_{2}+ ... +*a*_{n}*x*_{n}=*b*

*x*

_{1},

*x*

_{2}, ... ,

*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.