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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