Hyperplane

:A hyperplane is not to be confused with a hypersonic aircraft.

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In geometry, in one-dimensional space (such as a number line), a hyperplane is a point which divides the line in two. In two-dimensional space (such as the XY plane), a hyperplane is a line which divides the plane in two. In three-dimensional space, a hyperplane is a plane which divides the space in two. This concept can also be applied to four-dimensional space and beyond, where the dividing object is simply referred to as a hyperplane.

Related Topics:
Geometry - Point - Line - Plane

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The plus sign represents a point dividing the line in two.

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A point is a "hyperplane" in one-dimensional space.

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In four-dimensional space, a hyperplane is a three-dimensional subspace which divides the space in two. Note that, as above, the two parts on either side of the subspace^1 have zero distance between each other. This illustrates that even the most basic concepts of 4D space are difficult to visualize geometrically.

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In the general case, a hyperplane is a linear, affine, or projective subspace of codimension 1. In other words, a hyperplane is a higher-dimensional analog of a (two-dimensional) plane in three-dimensional space.

Related Topics:
Linear - Affine - Codimension

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Considering ordinary real linear or affine space, a hyperplane divides the space into two half-spaces, but in projective space or in any complex space it does not divide the space into parts. (For hyperplanes in spaces whose field of scalars is other than the real or complex numbers, such as a finite field, the concept of division into parts is usually meaningless.)

Related Topics:
Real - Half-space - Complex - Field - Finite field

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An affine hyperplane in n-dimensional space can be described by a non-degenerate linear equation of the following form:

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:a1x1 + a2x2 + ... + anxn = b.

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A linear hyperplane is similar but with b = 0.

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The two half-spaces defined by a hyperplane in n-dimensional space are:

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:a1x1 + a2x2 + ... + anxn ≤ b

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and

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:a1x1 + a2x2 + ... + anxn ≥ b.

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