# Hyperplane

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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* − 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* − 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 's is non-zero):

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

and

As an example, a point is a hyper plane 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.^{[1]} 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.

## 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 .^{[2]} The intersection of between P and H is defined to be a "face" of the polyhedron. The theory of polyhedron and the dimension of the faces are analyzed by the looking at these intersections involving hyper planes.

## See also

- Hypersurface
- Decision boundary
- Ham sandwich theorem
- Arrangement of hyperplanes
- Separating hyperplane theorem
- Supporting hyperplane theorem

## References

- 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 .