| Hyperplane |
Articles about Hyperplane |
Information AboutHyperplane |
| CATEGORIES ABOUT HYPERPLANE | |
| euclidean geometry | |
| affine geometry | |
| linear algebra | |
| projective geometry | |
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A hyperplane is a concept in Geometry . It is a generalization of the concept of a Plane . In a one-dimensional space (such as a 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 two Half-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. FORMAL DEFINITION In the general case, a hyperplane is an Affine Subspace of Codimension 1. 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: a Here, ''non-degenerate'' means that not all the ''a''''i'' are zero. If ''b''=0, one obtains a linear hyperplane, which goes through the origin of the space. The two half-spaces defined by a hyperplane in ''n''-dimensional space are: a and a NOTES The term realm has been advocated for a three-dimensional hyperplane in four-dimensional space, but this is not in common use. SEE ALSO |
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