 in geometry a hyperplane is a subspace of one dimension less than its ambient space. if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyperplanes are the 1-dimensional lines. this notion can be used in any general space in which the concept of the dimension of a subspace is defined. in different settings, the objects which are hyperplanes may have different properties. for instance, a hyperplane of an n-dimensional affine space is a flat subset with dimension n&nbsp;&minus;&nbsp;1. by its nature, it separates the space into two half spaces. but a hyperplane of an n-dimensional projective space does not have this property. == technical description == in geometry, a hyperplane of an n-dimensional space v is a subspace of dimension n&nbsp;&minus;&nbsp;1, or equivalently, of codimension&nbsp;1 in&nbsp;v. the space v may be a euclidean space or more generally an affine space, or a vector space or a projective space, and the notion of hyperplane varies correspondingly since the definition of subspace differs in these settings; in all cases however, any hyperplane can be given in coordinates as the solution of a single (due to the "codimension&nbsp;1" constraint) algebraic equation of degree&nbsp;1. if v is a vector space, one distinguishes "vector hyperplanes" (which are linear subspaces, and therefore must pass through the origin) and "affine hyperplanes" (which need not pass through the origin; they can be obtained by translation of a vector hyperplane). a hyperplane in a euclidean space separates that space into two half spaces, and defines a reflection that fixes the hyperplane and interchanges those two half spaces. ==special types of hyperplanes== several specific types of hyperplanes are defined with properties that are well suited for particular purposes. some of these specializations are described here. === affine hyperplanes === an affine hyperplane is an affine subspace of codimension 1 in an affine space. in cartesian coordinates, such a hyperplane can be described with a single linear equation of the following form (where at least one of the a_i's is non-zero): :a_1x_1 + a_2x_2 + \cdots + a_nx_n = b.\ in the case of a real affine space, in other words when the coordinates are real numbers, this affine space separates the space into two half-spaces, which are the connected components of the complement of the hyperplane, and are given by the [[inequalitie
Affine_geometry Euclidean_geometry Linear_algebra Projective_geometry