Hyperplane

A hyperplane is a concept in geometry. It is a generalization of the plane into a different number of dimensions.

A hyperplane of an n-dimensional space is a flat subset with dimension n − 1. By its nature, it separates the space into two half spaces.

Contents

Technical description

In geometry, a hyperplane of an n-dimensional space V is a "flat" subset of dimension n − 1, or equivalently, of codimension 1 in V; it may therefore be referred to as an (n − 1)-flat of 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; 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 (due to the "flat" constraint). If V is a vector space, one distinguishes "vector hyperplanes" (which are 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.

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 reflections 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.

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 %2B a_2x_2 %2B \cdots %2B 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 inequalities

a_1x_1 %2B a_2x_2 %2B \cdots %2B a_nx_n < b\

and

a_1x_1 %2B a_2x_2 %2B \cdots %2B a_nx_n > b.\

As an example, 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 linear subspace of codimension 1. Such a hyperplane is the solution of a single homogeneous linear equation.

Projective hyperplanes

Projective hyperplanes, are used in projective geometry. 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 real 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 in real projective space, the space essentially "wraps around" so that both sides of a lone hyperplane are connected to each other.

See also

References

External links