Hyperplane

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In geometry, as a plane has one less dimension than space, a hyperplane is a subspace of one dimension less than its ambient space.

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.

Technical description[edit]

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[edit]

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.

Special types of hyperplanes[edit]

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[edit]

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 + a_2x_2  + \cdots + 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 + a_2x_2  + \cdots + a_nx_n < b\

and

a_1x_1 + a_2x_2  + \cdots + a_nx_n > b.\

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[edit]

In a vector space, a vector hyperplane is a subspace of codimension 1, only possibly shifted from the origin by a vector. Such a hyperplane is the solution of a single linear equation.

Projective hyperplanes[edit]

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.

Support hyperplanes[edit]

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 H\cap P \neq \varnothing.[1] 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[edit]

References[edit]

  1. ^ Polytopes, Rings and K-Theory by Bruns-Gubeladze

External links[edit]