§ 19: Polarities

Definition : A polarity is an involutory correlation, or in other words: a correlation π whose square π2 is identity.

Note : For a polarity π we have: π(X) = l ↔ π(l) = X.
If π(X) = l we call X and l pole and polar line.
If the point Y lies on the polar line l of X, so π(X) = l, then π(l) = X, and then, because π(Y) goes through π(l), X lies on the polar line of Y.

Definition : We call two points conjugate if they are lying on each other's polar line, so each on the polar line of the other.
A point is called autopolar (selfconjugate) if it's lying on its own polar line.
We call two lines conjugate if each goes through the pole of the other.
A line is called autopolar (selfconjugate) if it goes through its own pole.

Note : The natural correlation πo is an example of a polarity without autopolar points or lines (see O75).
In the next few sections we'll see that with each non-degenerate conic we have a polarity whose autopolar points are the points on the conic.

Definition : We call triangle X1X2X3 a polar triangle if π(Xi) = XjXk for each permutation i-j-k of 1-2-3. See O76.

Proposition : The line connecting two autopolar points can't be autopolar.

Proof: Let P and Q be two distinct autopolar points on an autopolar line l (l = π(X) and X on l).
Then either XP or XQ. Suppose XQ.
Since Q lies on π(X), both X and Q lie on π(Q).So l = π(X) = π(Q).
Then X = Q. Contradiction.

Proposition : Any polarity π induces an involution or identity on every non-autopolar line (the pairs of this involution are pairs of conjugate points).

Proof: Let l be a non-autopolar line, say with pole L outside l.
The restriction of π to l is a projectivity from the pencil of points l to the pencil of lines L.
Then θ: X → π(X).l is a projectivity from l onto l.
If Y = θ(X), then Y = π(X).l, so X = π(Y).l, so X = θ(Y). So θ is an involution or identity.

O75 Study again the natural correlation in the sphere model of O6. Do autopolar points or lines exist?

O76 Prove there exists at most one polar triangle with given vertices X1 and X2. Under which conditions does such a triangle exist?

O77 Formulate the dual of each proposition in this section.

O78 If the line l has at least three autopolar ponts, prove all points on l are autopolar.