In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is one less than that of the ambient space. Two lower-dimensional examples of hyperplanes are one-dimensional lines in a plane and zero-dimensional points on a line.
Most commonly, the ambient space is n-dimensional Euclidean space, in which case the hyperplanes are the (n − 1)-dimensional flats, each of which separates the space into two half spaces.[1] A reflection across a hyperplane is a kind of motion (geometric transformation preserving distance between points), and the group of all motions is generated by the reflections. A convex polytope is the intersection of half-spaces.
In non-Euclidean geometry, the ambient space might be the n-dimensional sphere or hyperbolic space, or more generally a pseudo-Riemannian space form, and the hyperplanes are the hypersurfaces consisting of all geodesics through a point which are perpendicular to a specific normal geodesic.
In other kinds of ambient spaces, some properties from Euclidean space are no longer relevant. For example, in affine space, there is no concept of distance, so there are no reflections or motions. In a non-orientable space such as elliptic space or projective space, there is no concept of half-planes. In greatest generality, the notion of hyperplane is meaningful in any mathematical space in which the concept of the dimension of a subspace is defined.
The difference in dimension between a subspace and its ambient space is known as its codimension. A hyperplane has codimension 1.
^"Excerpt from Convex Analysis, by R.T. Rockafellar" (PDF). u.arizona.edu.
In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like...
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consists of a reflection across a hyperplane and a translation ("glide") in a direction parallel to that hyperplane, combined into a single transformation...
is replaced by a hyperplane. The problem of determining if a pair of sets is linearly separable and finding a separating hyperplane if they are, arises...
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(n − 1)-polytope in a (n − 1)-dimensional hyperplane. A point called apex is located outside the hyperplane and gets connected to all the vertices of...
consequence of the Cauchy–Schwarz inequality. The vector equation for a hyperplane in n {\displaystyle n} -dimensional Euclidean space R n {\displaystyle...
space of dimension n is a set of n + 1 points that are not contained in a hyperplane. An affine basis define barycentric coordinates for every point. Many...
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}}={\frac {dw}{d\tau }},} then they share the same simultaneous hyperplane. This hyperplane exists mathematically, but physical relations in relativity involve...
which is a hyperplane passing through the origin, and its norm. Geometrically thus, covariant vectors should be viewed as a set of hyperplanes, with spacing...
a linear transformation that describes a reflection about a plane or hyperplane containing the origin. The Householder transformation was used in a 1958...