Duality

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Other related sources of information:

Duality, at Wikipedia
Duality in mathematics, at Wikipedia
Duality in projective geometry, at Wikipedia

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In projective geometry
a duality is an involutory transformation
that changes the relations
between a \, s_{tructure} \,
and its dual structure, often denoted \, {s_{tructure}}^* ,
which is complementary to the original structure
with respect to some context
that provides a background for the duality.

Within the context of a projective plane \, \mathbb{P}^{\, 2} we have:
The dual of a point is a line and the dual of a line is a point,
which we can represent as:

\, [ \,\, {p_{oint}}^* \, ]_{\, \mathbb{P}^{\, 2}} \, = \, \left< \,\, l_{ine} \, \right>_{\, \mathbb{P}^{\, 2}}

\, [ \,\, {l_{ine}}^* \, ]_{\, \mathbb{P}^{\, 2}} \, = \, \left< \,\, p_{oint} \, \right>_{\, \mathbb{P}^{\, 2}}

Within the context of a projective \, 3 -space \, \mathbb{P}^{\, 3} we have:
The dual of a point is a plane and the dual of a plane is a point,
which we can represent as:

\, [ \,\, {p_{oint}}^* \, ]_{\, \mathbb{P}^{\, 3}} \, = \, \left< \,\, p_{lane} \, \right>_{\, \mathbb{P}^{\, 3}}

\, [ \,\, {p_{lane}}^* \, ]_{\, \mathbb{P}^{\, 3}} \, = \, \left< \,\, p_{oint} \, \right>_{\, \mathbb{P}^{\, 3}}

Within the context of a projective \, n-space \, \mathbb{P}^{\, n} we have:
The dual of a point is a hyperplane and the dual of a hyperplane is a point,
which we can represent as:

\, [ \,\, {p_{oint}}^* \, ]_{\, \mathbb{P}^{\, n}} \, = \, \left< \,\, h_{yperplane} \, \right>_{\, \mathbb{P}^{\, n}}

\, [ \,\, {h_{yperplane}}^* \, ]_{\, \mathbb{P}^{\, n}} \, = \, \left< \,\, p_{oint} \, \right>_{\, \mathbb{P}^{\, n}}

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