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A hyperplane is a concept in geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
. It is a higher-dimensional generalization of the concepts of a line in the plane and a plane in 3-dimensional space. The most familiar kinds of hyperplane are affine
Affine geometry

In mathematics affine geometry is the study of geometric properties which remain unchanged by affine transformations, i.e. non-singular linear transformations and Translation s....
 and linear
Linear algebra

Linear algebra is the branch of mathematics concerned with the study of Euclidean vectors, vector spaces , linear maps , and system of linear equations....
 hyperplanes; less familiar is the projective
Projective geometry

In mathematics projective geometry is the study of geometric properties which are invariant under projective transformations. The field of projective geometry is itself divided into many subfields, two examples of which are projective algebraic geometry and projective differential geometry ....
 hyperplane.

In a one-dimensional space (a straight line), a hyperplane is a point
Point (geometry)

In geometry, topology and related branches of mathematics a spatial point describes a specific object within a given space that consists of neither volume, area, length, nor any other higher dimensional analogue....
; it divides a line into two rays. In two-dimensional space (such as the xy plane), a hyperplane is a line
Line (mathematics)

In geometry, a line is a Curvature curve. When geometry is used to model the real world, lines are used to represent straight objects with negligible width and height....
; it divides the plane into two half-planes.






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A hyperplane is a concept in geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
. It is a higher-dimensional generalization of the concepts of a line in the plane and a plane in 3-dimensional space. The most familiar kinds of hyperplane are affine
Affine geometry

In mathematics affine geometry is the study of geometric properties which remain unchanged by affine transformations, i.e. non-singular linear transformations and Translation s....
 and linear
Linear algebra

Linear algebra is the branch of mathematics concerned with the study of Euclidean vectors, vector spaces , linear maps , and system of linear equations....
 hyperplanes; less familiar is the projective
Projective geometry

In mathematics projective geometry is the study of geometric properties which are invariant under projective transformations. The field of projective geometry is itself divided into many subfields, two examples of which are projective algebraic geometry and projective differential geometry ....
 hyperplane.

In a one-dimensional space (a straight line), a hyperplane is a point
Point (geometry)

In geometry, topology and related branches of mathematics a spatial point describes a specific object within a given space that consists of neither volume, area, length, nor any other higher dimensional analogue....
; it divides a line into two rays. In two-dimensional space (such as the xy plane), a hyperplane is a line
Line (mathematics)

In geometry, a line is a Curvature curve. When geometry is used to model the real world, lines are used to represent straight objects with negligible width and height....
; it divides the plane into two half-planes. In three-dimensional space, a hyperplane is an ordinary plane
Plane (mathematics)

In mathematics, a plane is a curvature surface. Planes can arise as subspaces of some higher dimensional space, as with the walls of a room, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry....
; it divides the space into two half-space
Half-space

In geometry, a half-space is either of the two parts into which a plane divides the three-dimensional space. More generally, a half-space is either of the two parts into which a hyperplane divides an affine space....
s. This concept can also be applied to four-dimensional space and beyond, where the dividing object is simply referred to as a "hyperplane".

Affine hyperplanes

In the general case, an affine hyperplane is an affine subspace
Affine space

In mathematics, an affine space is a geometric structure that generalizes the affine geometry properties of Euclidean space. It can be thought of informally as a vector space where one has forgotten which point is the origin....
 of codimension
Codimension

In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, and more generally to submanifolds in manifolds, and suitable subsets of algebraic varieties....
 1 in an affine geometry
Affine geometry

In mathematics affine geometry is the study of geometric properties which remain unchanged by affine transformations, i.e. non-singular linear transformations and Translation s....
. In other words, a hyperplane is a higher-dimensional analog of a (two-dimensional) plane in three-dimensional space.

An affine hyperplane in n-dimensional space with coordinates in a field
Field (mathematics)

In abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication and division , satisfying certain axioms....
 K can be described by a non-degenerate linear equation
Linear equation

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable.Linear equations can have one or more variables....
 of the following form:

a1x1 + a2x2 + ... + anxn = b.


Here, non-degenerate means that not all the ai are zero. If b=0, one obtains a linear or homogeneous hyperplane, which goes through the origin of the coordinate system.

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

a1x1 + a2x2 + ... + anxn = b


and

a1x1 + a2x2 + ... + anxn = b.


Linear hyperplanes


In linear algebra the term "hyperplane" is used in a more limited way. A hyperplane in a vector space
Vector space

File:Vector addition ans scaling.pngA vector space is a mathematical structure formed by a collection of vectors: objects that may be Vector addition together and Scalar multiplication by numbers, called scalar s in this context....
 is a vector subspace (or "linear subspace") whose dimension is 1 less than that of the whole vector space. These hyperplanes are the affine hyperplanes that contain the origin of coordinates.

Projective hyperplanes


There are also projective hyperplanes, in projective geometry. Projective geometry can be viewed as affine geometry with vanishing point
Vanishing point

A vanishing point is a point in a Perspective drawing to which parallel lines appear to converge. The number and placement of the vanishing points determines which perspective technique is being used....
s (points at infinity) added. An affine hyperplane together with the associated points at infinity forms a projective hyperplane. There is one other projective hyperplane: the set of all points at infinity, called the infinite or ideal hyperplane.

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
    Hypersurface

    In geometry, a hypersurface is a generalization of the concept of hyperplane. Suppose an enveloping manifold M has n dimensions; then any submanifold of M of n − 1 dimensions is a hypersurface....
  • decision boundary
    Decision boundary

    In a statistical classification problem with two classes, a decision boundary or decision surface is a hypersurface that partitions the underlying vector space into two sets, one for each class....
  • ham sandwich theorem
    Ham sandwich theorem

    In measure theory, a branch of mathematics, the ham sandwich theorem, also called the Stone?Tukey theorem after Arthur Harold Stone and John Tukey, states that given n "objects" in n-dimensional space, it is possible to divide all of them in half with a single -dimensional hyperplane....
  • arrangement of hyperplanes
    Arrangement of hyperplanes

    In geometry and combinatorics, an arrangement of hyperplanes is a finite set A of hyperplanes in a linear space, affine geometry, or projective geometry space S....