# Hyperplane

In geometry, a hyperplane is a subspace whose dimension is one less than that of its ambient space. For example, 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 and sufficient condition for S to be a hyperplane in X is for S to have codimension one in X.

## 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}

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: ${\displaystyle \pm {\hat {n}}}$ . In particular, if we consider ${\displaystyle \mathbb {R} ^{n+1}}$  equipped with the conventional inner product (dot product), then one can define the affine subspace with normal vector ${\displaystyle {\hat {n}}}$  and origin translation ${\displaystyle {\tilde {b}}\in \mathbb {R} ^{n+1}}$  as the set of all ${\displaystyle x\in \mathbb {R} ^{n+1}}$  such that ${\displaystyle {\hat {n}}\cdot (x-{\tilde {b}})=0}$ .

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.

The datapoint and its predicted value via a linear model is a hyperplane.

## 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 looking at these intersections involving hyperplanes.