[alogo] Duality

Usually by Duality we mean a bijection between a projective plane P2 and its dual, which is defined as the set of p-lines (projective lines) of P2. This set has again the structure of a projective plane. To illustrate the ideas consider the projective plane P2, identified with the set of e-lines (euclidean lines) through the origin of R3. Such an e-line is represented by [x], where x=(x1,x2,x3) is a non-zero point of R3. The projective lines of P2 (call them p-lines) consist of all e-lines of R3 lying in an e-plane (euclidean plane) through the origin. Hence the points [x] of this p-line satisfy an equation of the form a1*x1+a2*x2+a3*x3=0. Denote the set of non-zero multiples of a=(a1,a2,a3) by [a]. Each [a] represents a p-line of P2 and the set of all these [a]'s builds a copy of P2 called the dual of P2 and denoted by P2*.

[0_0] [0_1]

The basic relation between points [x] of P2 and points [a] of P2* is that of coincidence, expressed by the equality a1*x1+a2*x2+a3*x3=0. The equation expressing that p-point [x] is on the p-line [a], or equivalently, that the p-line [a] passes through the p-point [x].
Relation a1*x1+a2*x2+a3*x3=0, can be read from two sides.
First, fixing [x] and considering all [a]'s satisfying this relation.
Second, fixing [a] and considering all [x]'s satisfying this relation.
In the first case all [a]'s define a p-line in P2* consisting of the bundle of p-lines passing through [x]. We denote this bundle by [x]*. In terms of our euclidean eyes this idendifies line [x] with all the planes containing this line. The map [x]-->[x]* establishes a natural bijection between the projective spaces P2 and (P2*)* (though there is no intrisec bijection between P2 and P2*).
In the second case all [x]'s define a p-line of P2 consisting of all p-points of P2 contained in the line [a]. We could denote it by [a]*, but this is identified with [a]. Abstractly we have the relation [x]** = [x].
The important thing is that the set of lines of P2 has the same structure as P2 itself. Further the lines of P2* are identified with [x]* i.e. the bundle of lines through point [x]. Thus the two conditions become equivalent:
(1) line [a] = join([x],[y]), for points [x], [y] of P2 and [a] a line of P2 (equivalently a point of P2*).
(2) point [a] = inter([x]*,[y]*) , [x]*, [y]* being lines of P2* and [a] being a point of P2*.


Thus, in theorems of projective geometry, intechanging the words point<--->line and verbs join<--->intersect we get dual new theorems. The new theorems being valid in P2*, which is isomorphic to P2. We say then proof by duality and prove only one version of the two dual theorems.
A typical case is the theorem of Desargues (see Desargues.html ).

Theorem
I and II below are equivalent.
(I) For triangles [x][y][z], [x'][y'][z'], lines [a]=join([x],[x']), [b]=join([y][y']), [c]=join([z][z']) are concurrent, i.e. there is a point [w] lying on all three lines [a], [b], [c].
(II) Intersection points of sides [p]=inter([xy],[x'y']), [q]=inter([yz],[y'z']), [r]=inter([zx],[z'x']) lie on the same line [e].

Here [xy] denotes the line [f]=join([x],[y]), [yz] denotes [g]=join([y],[z]), etc..
Assume we proved that I => II. Then to prove II => I take the dual of II:
[p]([xy],[x'y']) reads: line [p]* joining points [f] and [f'] (in the dual space P2*).
[q]([yz],[y'z']) reads: line [q]* joining points [g] and [g'] (in the dual space P2*).
[r]([zx],[z'x']) reads: line [r]* joining points [h] and [h'] (in the dual space P2*).
lie on the same line [e] reads: [p]*,[q]*,[r]* intersect at [e] (again in P2*).
Thus, II translates:
For triangles [f][g][h], [f'][g'][h'] of P2* lines join([f],[f']), join([g],[g']), join([h],[h']) are concurrent. But this is just I stated for the dual space P2*. By the proven part of the theorem follows that:
[u]([fg],[f'g']) etc. are on the same line [o]* (of P2*). But [fg] is [x], [f'g'] is [x'], [u] is line [xx'] etc.. Hence [u], [v], [w] being on the same line [o]* means that line [xx'] passes through [o] and similar does [yy'] and [zz'].
Thus, the dual (II') of (II) is (I) reformulated in the projective space P2*. Also the dual (I') of (I) is (II) reformulated in P2*. Discarding the proper nature of the projective spaces (i.e. considering P2 and P2* to be abstractly the same thing) we see that {I=>II} and {II=>I} expresses the same proposition in two different projective spaces.
A similar situation arises in the theorem of Pappus, which is self-dual. i.e. the dual expresses the same figure as the original (see PappusSelfDual.html ). Another instance of duality is that between the theorems of Brianchon and Pascal.

See Also

Brianchon.html
Desargues.html
Menelaus.html
MenelausApp.html
PappusLines.html
PappusSelfDual.html
Pascal.html

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