Hyperplane
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- A hyperplane is not to be confused with a hypersonic aircraft.
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 kind of hyperplane is an affine hyperplane; that is the kind described here.
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.
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[edit] Formal definition
In the general case, an affine hyperplane is an affine subspace of codimension 1 in an affine space. 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 can be described by a non-degenerate linear equation 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 hyperplane, which goes through the origin of the coordinate system.
The two half-spaces defined by a hyperplane in n-dimensional space are:
- a1x1 + a2x2 + ... + anxn ≤ b
and
- a1x1 + a2x2 + ... + anxn ≥ b.
[edit] Projective hyperplanes
There are also projective hyperplanes in projective geometry. This can be viewed as affine geometry with vanishing points (points at infinity) added. An affine hyperplane together with the associated points at infinity form a projective hyperplane. There is one other projective hyperplane: the set of all points at infinity.
[edit] Notes
The term realm has been proposed for a three-dimensional hyperplane in four-dimensional space, but it is used rarely, if ever.
