Duality (projective geometry)

In projective geometry , dualities serve to swap the role of points and straight lines in a projective plane. This is useful because by swapping points and lines, difficult problems can often be transformed into simpler or already solved problems.

definition

Let it be the projective plane of a 3-dimensional vector space and the set of straight lines in . ${\displaystyle P=P(V)}$ ${\displaystyle V}$ ${\displaystyle P^{*}}$ ${\displaystyle P}$

A duality is a homeomorphism that maps collinear points into coincident straight lines . ${\displaystyle P\to P^{*}}$

example

Each scalar product is defined by means of ${\displaystyle \langle .,.\rangle }$ ${\displaystyle V}$

${\displaystyle \left[v\right]\to \left[v\right]^{\perp }=\left\{\left[w\right]\in P(V):\langle v,w\rangle =0\right\}}$

a bijection

${\displaystyle P\to P^{*}}$.

Such dualities defined by a scalar product are called polarities .

Principle of duality

For each theorem of projective geometry there is a dual theorem, which is obtained from the fact that in the formulation of the theorem, "straight lines" through "points" and "points" through "straight lines", as well as "intersection of two straight lines" through "straight lines through two Points ”and“ Straight lines through two points ”can be replaced by“ Intersection of two straight lines ”.

Examples

• The set of Pascal is dual to the set of Brianchon .
• The Ceva's theorem is dual to the Menelaus theorem .
• The set of Desargues is dual to itself.

Web links

• Duality Principle (MathWorld)

Individual evidence

1. ^ Dickinson: The Theorems of Ceva and Menelaos and the Principle of Duality . In: The Mathematical Gazette , 1964, JSTOR 3611708 .