Editing Hyperplane (section)

==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 space|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 <math>a_i</math>s is non-zero and <math>b</math> is an arbitrary constant):

:<math>a_1x_1 + a_2x_2  + \cdots + a_nx_n = b.\ </math>

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 component (topology)|connected component]]s of the [[complement (set theory)|complement]] of the hyperplane, and are given by the [[inequality (mathematics)|inequalities]]

:<math>a_1x_1 + a_2x_2  + \cdots + a_nx_n < b\ </math>

and

:<math>a_1x_1 + a_2x_2  + \cdots + a_nx_n > b.\ </math>

As an example, a point is a hyperplane in 1-dimensional space, a line is a hyperplane in 2-dimensional space, and a plane is a hyperplane in 3-dimensional space. A line in 3-dimensional space is not a hyperplane, and does not separate the space into two parts (the complement of such a line is connected).

Any hyperplane of a Euclidean space has exactly two unit normal vectors: <math>\pm\hat{n}</math>. In particular, if we consider <math>\mathbb{R}^{n+1}</math> equipped with the conventional inner product ([[dot product]]), then one can define the affine subspace with normal vector <math>\hat{n}</math> and origin translation <math>\tilde{b} \in \mathbb{R}^{n+1}</math> as the set of all <math>x \in \mathbb{R}^{n+1}</math> such that <math>\hat{n} \cdot (x-\tilde{b})=0</math>.

Affine hyperplanes are used to define decision boundaries in many [[machine learning]] algorithms such as linear-combination (oblique) [[Decision tree learning|decision trees]], and [[perceptron]]s.

=== Vector hyperplanes ===

In a vector space, a vector hyperplane is a [[Linear subspace|subspace]] of codimension&nbsp;1, only possibly shifted from the origin by a vector, in which case it is referred to as a [[flat (geometry)|flat]]. Such a hyperplane is the solution of a single [[linear equation]].

===Projective hyperplanes===
'''Projective hyperplanes''', are used in [[projective geometry]].  A [[Projective geometry#Projective subspace|projective subspace]] is a set of points with the property that for any two points of the set, all the points on the line determined by the two points are contained in the set.<ref>{{citation|first1=Albrecht|last1=Beutelspacher|first2=Ute|last2=Rosenbaum|title=Projective Geometry: From Foundations to Applications|year=1998|publisher=Cambridge University Press|isbn=9780521483643|page=10}}</ref> Projective geometry can be viewed as [[affine geometry]] with [[vanishing point]]s (points at infinity) added.  An affine hyperplane together with the associated points at infinity forms a projective hyperplane.  One special case of a projective hyperplane is the '''infinite''' or '''ideal hyperplane''', which is defined with the set of all points at infinity.

In  projective space, a hyperplane does not divide the space into two parts; rather, it takes two hyperplanes to separate points and divide up the space.  The reason for this is that the space essentially "wraps around" so that both sides of a lone hyperplane are connected to each other.