Hyperplane

A hyperplane is a concept in geometry. It is a generalization of the concept of a plane into a different number of dimensions. Analogous with a plane which defines a two-dimensional subspace in a three-dimensional space, a hyperplane defines a k-dimensional subspace within an n-dimensional space, where k<n. A line, for example, is a one-dimensional hyperplane in a space with any number of dimensions. High dimensional hyperplanes are difficult to visualize, but they share many mathematical properties in common with regular lines and planes.

Definition

A k-dimensional hyperplane (also called a k-hyperplane) can be defined in n-dimensional space by k+1 points. For example, in 3-space, a line is defined by two points, and a plane is defined by three points. (Each of the points is specified as a vector of n coordinate values to indicate a position in n-dimensional space, and no three of the points can be colinear.)

Equivalently, a k-dimensional hyperplane can be defined with an origin point and a set of k orthonormal basis vectors. This form is commonly used to simplify calculations related to hyperplanes. (It may also be defined with k non-orthonormal basis vectors, but this does not have all the nice mathematical properties.)

Also equivalently, a k-dimensional hyperplane can be defined in n-dimensional space by a set of (n-k) non-degenerate linear equations. (For example, it takes one linear equation to define a line in 2-space, it takes two linear equations to define a line in 3-space, and it takes one linear equation to define a plane in 3-space.) Sample points that lie on a hyperplane can be computed from these equations, and vice versa.

Computations

All of the techniques in this section are applicable to lines, planes, and hyperplanes.

Distance to a point

The distance between a point and a hyperplane can be computed in two steps: First, project the point onto the hyperplane. Second, compute the distance between the point and the projected point.

Projecting a point onto a hyperplane

A point may be projected onto a hyperplane in three steps as follows: First, select an origin on the hyperplane, and compute a set of orthonormal basis vectors that lie on the hyperplane. Second, project the point onto each of the orthonormal basis vectors. Third, add each of these projections to the origin. The result is the projection of the point onto the hyperplane.

Computing orthonormal basis vectors on a hyperplane

A set of orthonormal basis vectors that lie on a hyperplane can be computed in two steps as follows: First, subtract the first point used to define the hyperplane from each of the remaining points to obtain a set of vectors that lie on the hyperplane. (Thus, the first point serves as the origin for these vectors.) Second, use the Gram–Schmidt process to orthogonalize the vectors.

Computing the dihedral angle between two hyperplanes

The dihedral angle between two hyperplanes can be computed as follows: First, select a random vector that is not parallel to either of the hyperplanes. Make a copy of the random vector. Using one copy of the random vector, subtract the component that projects onto each of the orthonormal basis vectors of the first hyperplane. Using the other copy, subtract the component that projects onto each of the orthonormal basis vectors of the other hyperplane. (The two vectors are now orthogonal to the two hyperplanes, and are otherwise correlated since they started as the same vector.) The angle between the two orthogonal vectors is the dihedral angle between the two hyperplanes. This can be computed as the arc-cosine of the linear correlation of the two vectors.

Principle Components

Given a collection of points, it is common to use the mean as an origin and the first k-principle components as a set of orthonormal basis vectors to define a hyperplane. A hyperplane defined in this manner has the smallest sum-squared distance with the points in the collection of any hyperplane that passes through that origin. This technique is used in local neighborhoods to estimate the tangent space of a manifold from points that are sampled from the manifold.

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 geometry. In other words, it is a hyperplane where n-k=1.

An affine hyperplane can be described with a single linear equation of the following form:

a₁x₁ + a₂x₂ + ... + a_nx_n = b.

All affine hyperplanes have exactly two possible normal vectors. Non-affine hyperplanes have an infinite number of normal vectors. (For example, consider the normal vectors of a line in 2-space, and then consider the normal vectors of a line in 3-space. There are two, and infinite, respectively.) Affine hyperplanes also divide the space into exactly two parts, whereas non-affine hyperplanes do not divide the space.

The two half-spaces defined by an affine hyperplane in n-dimensional space with real-number coordinates are:

a₁x₁ + a₂x₂ + ... + a_nx_n ≤ b

and

a₁x₁ + a₂x₂ + ... + a_nx_n ≥ b.

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

Linear hyperplanes

A linear hyperplane is one that passes through the origin. Such hyperplanes can be represented with a (n-k) by n matrix that stores the coefficients of the equations that define the hyperplane, or equivalently, the basis vectors that lie on the hyperplane. To fully define a hyperplane that does not pass through the origin, an additional (n-k) dimensional vector is required to store the values of b for each 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.

See also

• hypersurface
• decision boundary
• ham sandwich theorem
• arrangement of hyperplanes

Notes

• Hyperplanes in complex affine space do not divide the space into two parts. For this property, the coordinate field has to be an ordered field.
• The term realm has been proposed for a three-dimensional hyperplane in four-dimensional space, but it is used rarely, if ever.