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hyperplane
In geometry a hyperplane is a subspace of one dimension less than its ambient space. If a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyperplanes are the 1-dimensional lines. This notion can be used in any general space in which the concept of the dimension of a subspace is defined.
In different settings, the objects which are hyperplanes may have different properties. For instance, a hyperplane of an ''n''-dimensional affine space is a flat subset with dimension ''n'' − 1. By its nature, it separates the space into two half spaces. But a hyperplane of an ''n''-dimensional projective space does not have this property.
== Technical description ==
In geometry, a hyperplane of an ''n''-dimensional space ''V'' is a subspace of dimension ''n'' − 1, or equivalently, of codimension 1 in ''V''. The space ''V'' may be a Euclidean space or more generally an affine space, or a vector space or a projective space, and the notion of hyperplane varies correspondingly since the definition of subspace differs in these settings; in all cases however, any hyperplane can be given in coordinates as the solution of a single (due to the "codimension 1" constraint) algebraic equation of degree 1.
If ''V'' is a vector space, one distinguishes "vector hyperplanes" (which are linear subspaces, and therefore must pass through the origin) and "affine hyperplanes" (which need not pass through the origin; they can be obtained by translation of a vector hyperplane). A hyperplane in a Euclidean space separates that space into two half spaces, and defines a reflection that fixes the hyperplane and interchanges those two half spaces.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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