# Hyperplane

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In geometry, a **hyperplane** is a subspace whose dimension is one less than that of 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, hyperplanes may have different properties. For instance, a hyperplane of an *n*-dimensional affine space is a flat subset with dimension *n* − 1^{[1]} and it separates the space into two half spaces. While a hyperplane of an *n*-dimensional projective space does not have this property.

The difference in dimension between a subspace S and its ambient space X is known as the codimension of S with respect to X. Therefore, a necessary condition for S to be a hyperplane in X is for S to have codimension one in X.

## Contents

## Technical description

In geometry, a **hyperplane** of an *n*-dimensional space *V* is a subspace of dimension *n* − 1, or equivalently, of codimension 1 in *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 1" constraint) algebraic equation of degree 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 \({\displaystyle a_{i}}\)'s is non-zero and \({\displaystyle b}\) is an arbitrary constant):

- \({\displaystyle a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{n}x_{n}=b.\ }\)

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 inequalities

- \({\displaystyle a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{n}x_{n}<b\ }\)

and

- \({\displaystyle a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{n}x_{n}>b.\ }\)

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.

Affine hyperplanes are used to define decision boundaries in many machine learning algorithms such as linear-combination (oblique) decision trees, and perceptrons.

### Vector hyperplanes

In a vector space, a vector hyperplane is a subspace of codimension 1, only possibly shifted from the origin by a vector, in which case it is referred to as a flat. Such a hyperplane is the solution of a single linear equation.

### Projective hyperplanes

**Projective hyperplanes**, are used in projective geometry. A 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.^{[2]} Projective geometry can be viewed as affine geometry with vanishing points (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.

## Applications

In convex geometry, two disjoint convex sets in n-dimensional Euclidean space are separated by a hyperplane, a result called the hyperplane separation theorem.

In machine learning, hyperplanes are a key tool to create support vector machines for such tasks as computer vision and natural language processing.

## Dihedral angles

The dihedral angle between two non-parallel hyperplanes of a Euclidean space is the angle between the corresponding normal vectors. The product of the transformations in the two hyperplanes is a rotation whose axis is the subspace of codimension 2 obtained by intersecting the hyperplanes, and whose angle is twice the angle between the hyperplanes.

### Support hyperplanes

A hyperplane H is called a "support" hyperplane of the polyhedron P if P is contained in one of the two closed half-spaces bounded by H and \({\displaystyle H\cap P\neq \varnothing }\).^{[3]} The intersection of P and H is defined to be a "face" of the polyhedron. The theory of polyhedra and the dimension of the faces are analyzed by the looking at these intersections involving hyperplanes.

## See also

## References

**^**"Excerpt from Convex Analysis, by R.T. Rockafellar" (PDF).*u.arizona.edu*.**^**Beutelspacher, Albrecht; Rosenbaum, Ute (1998),*Projective Geometry: From Foundations to Applications*, Cambridge University Press, p. 10, ISBN 9780521483643**^**Polytopes, Rings and K-Theory by Bruns-Gubeladze

- Binmore, Ken G. (1980).
*The Foundations of Topological Analysis: A Straightforward Introduction: Book 2 Topological Ideas*. Cambridge University Press. p. 13. ISBN 0-521-29930-6. - Charles W. Curtis (1968)
*Linear Algebra*, page 62, Allyn & Bacon, Boston. - Heinrich Guggenheimer (1977)
*Applicable Geometry*, page 7, Krieger, Huntington ISBN 0-88275-368-1 . - Victor V. Prasolov & VM Tikhomirov (1997,2001)
*Geometry*, page 22, volume 200 in*Translations of Mathematical Monographs*, American Mathematical Society, Providence ISBN 0-8218-2038-9 .

## External links

Look up in Wiktionary, the free dictionary.hyperplane |

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