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 + ... + anxn = b
This equation reduces the number of
degrees of freedom of the point (
x1,
x2, ... ,
xn) 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.