- 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.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-plane s. In three-dimensional space, a hyperplane is an ordinary plane; it divides the space into twohalf-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 of
codimension 1 in anaffine 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::"a"1"x"1 + "a"2"x"2 + ... + "a""n""x""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:
:"a"1"x"1 + "a"2"x"2 + ... + "a""n""x""n" ≤ "b"
and
:"a"1"x"1 + "a"2"x"2 + ... + "a""n""x""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 point 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.
ee 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 anordered field .
* The term realm has been proposed for a three-dimensional hyperplane in four-dimensional space, but it is used rarely, if ever.
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