Editing Hyperplane (section)

{{short description|Subspace of n-space whose dimension is (n-1)}}
{{Use American English|date = March 2019}}
[[File:Intersecting_planes.svg|thumb|Two intersecting planes: Two-dimensional planes are the hyperplanes in three-dimensional space.]]

In [[geometry]], a '''hyperplane''' is a generalization of a [[plane (geometry)|two-dimensional plane]] in [[three-dimensional space]] to [[space (mathematics)|mathematical spaces]] of arbitrary [[dimension]]. Like a [[Euclidean planes in three-dimensional space|plane in space]], a hyperplane is a [[flat space|flat]] [[hypersurface]], a [[subspace (mathematics)|subspace]] whose [[dimension]] is one less than that of the [[ambient space]]. Two lower-dimensional examples of hyperplanes are [[one-dimensional]] [[line (geometry)|lines]] in a plane and [[zero-dimensional]] [[point (geometry)|points]] on a line.

Most commonly, the ambient space is {{mvar|n}}-dimensional [[Euclidean space]], in which case the hyperplanes are the {{math|(''n''&thinsp;−&thinsp;1)}}-dimensional [[flat (geometry)|"flats"]], each of which separates the space into two [[half-space (geometry)|half space]]s.<ref>{{Cite web|url=http://www.u.arizona.edu/~mwalker/econ519/RockafellarExcerpt.pdf|title=Excerpt from Convex Analysis, by R.T. Rockafellar|website=u.arizona.edu}}</ref> A [[reflection (geometry)|reflection]] across a hyperplane is a kind of [[Motion (geometry)|motion]] ([[geometric transformation]] preserving [[distance (mathematics)|distance]] between points), and the [[group (mathematics)|group]] of all motions is [[Generating set of a group|generated]] by the reflections. A [[convex polytope]] is the [[intersection (set theory)|intersection]] of half-spaces.

In [[non-Euclidean geometry]], the ambient space might be the [[n-sphere|{{mvar|n}}-dimensional sphere]] or [[hyperbolic space]], or more generally a [[pseudo-Riemannian manifold|pseudo-Riemannian]] [[space form]], and the hyperplanes are the hypersurfaces consisting of all [[geodesic]]s through a point which are [[perpendicular]] to a specific [[normal (geometry)|normal]] geodesic.

In other kinds of ambient spaces, some properties from Euclidean space are no longer relevant. For example, in [[affine space]], there is no concept of distance, so there are no reflections or motions. In a [[orientability|non-orientable]] space such as [[elliptic space]] or [[projective space]], there is no concept of half-planes. In greatest generality, the notion of hyperplane is meaningful in any mathematical space in which the concept of the dimension of a [[Topological subspace|subspace]] is defined.

The difference in dimension between a subspace and its ambient space is known as its ''[[codimension]]''. A hyperplane has codimension {{math|1}}.