PROJECTIVE GEOMETRY COURSE

§ 19: *Polarities*

*Definition :* A polarity is an involutory correlation, or in other words: a correlation π whose square π^{2} 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 π_{o} 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 *X*_{1}*X*_{2}*X*_{3} a *polar triangle* if π(*X*_{i}) = *X*_{j}*X*_{k}
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 *X* ≠ *P* or *X* ≠ *Q*. Suppose *X* ≠ *Q*.

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 *X*_{1} and *X*_{2}. 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.