Hyperplane

A hyperplane is a concept in geometry. It is a higher-dimensional generalization of the concepts of a line in Euclidean plane geometry and a plane in 3-dimensional Euclidean geometry. The most familiar kinds of hyperplane are affine and linear hyperplanes; less familiar is the projective hyperplane. Think of affine space as a simplified kind of Euclidean space, with real-number coordinates. Then, in a one-dimensional space (a straight line), a hyperplane is a point; it divides a line into two rays. In two-dimensional space (such as the xy plane), a hyperplane is a line; it divides the plane into two half-planes. In three-dimensional space, a hyperplane is an ordinary plane; it divides the space into two half-spaces. 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 of codimension 1 in an affine geometry. 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 K can be described by a non-degenerate linear equation of the following form:

a₁x₁ + a₂x₂ + ... + a_nx_n = b.

Here, non-degenerate means that not all the a_i 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 + ... + a_nx_n ≤ b

and

a₁x₁ + a₂x₂ + ... + a_nx_n ≥ b.

Linear hyperplanes

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

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.

See also

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