PROJECTIVE GEOMETRY COURSE

§ 14: *Involutions*

*Definition :* We call a non-trivial projectivity φ from a line *l* onto itself (or from a pencil *L* onto itself) *involution* if φ^{2} is identity.

*Proposition :* If a projectivity φ from *l* onto itself (or from *L* onto itself) interchanges two points (lines), then φ is an involution.

*Proof :* Suppose that φ interchanges the points *A* and *B*, and let *X* be a third point on *l*. Then (*A*, *B*; *X*, φ(*X*)) =
(φ(*A*), φ(*B*); φ(*X*), φ^{2}(*X*)) = (*B*, *A*; φ(*X*), φ^{2}(*X*)).

We also have (*A*, *B*; *X*, φ(*X*)) = (*B*, *A*; φ(*X*), *X*). So according to *O38* we must have φ^{2}(*X*) = *X*.

*Remark :* We call {*X*, φ(*X*)} a *pair* of the involution.

*Proposition :* If an involution φ has a fixed point *P* (invariant line *p*), then φ has another fixed point *R* ≠ *P* (invariant line *r* ≠ *p*),
and the pairs of φ separate {*P*,*R*} harmonically.

*Proof :* Let {*X*,φ(*X*)} be a pair of φ with *X* ≠ φ(*X*). Let *R* be the fourth harmonic with *P* and {*X*,φ(*X*)}.
We have to prove that
φ(*R*) = *R*. (Then it follows that each *X* yields the same *R*, because a non-trivial projectivity can have at most two fixed points.)

Indeed: (*P*, *R*; *X*, φ(*X*)) = -1, so (*P*, *R*; *X*, φ(*X*)) = (*R*, *P*; *X*, φ(*X*)). Also
(*P*, *R*; *X*, φ(*X*)) = (φ(*P*), φ(*R*); φ(*X*), φ^{2}(*X*)) = (*P*, φ(*R*); φ(*X*), *X*) =
(φ(*R*), *P*; *X*, φ(*X*)).

According to *O38*, we find φ(*R*) = *R*.

*Remark :* So we can construct with each point on *l* (each line through *L*) the image point (the image line) if the fixed points (invariant lines) are given (see *O49*).

*Remark :* The proposition implies that there doesn't exist any parabolic involution.

The involution with pairs {*A*,*A'*} and {*B*,*B'*} is elliptic if {*A*,*A'*} and {*B*,*B'*} separate. Indeed, suppose that this involution is hyperbolic
with fixed points *P* and *Q*.

Because of the harmonic positions with a hyperbolic involution we would have:

*x _{2}*/(

A contradiction follows, because the first number is smaller than the third (check this) and the second is greater than the fourth.

*O55* Determine the matrix of the involution with pairs {(4,-5),(1,2)} and {(5,-4),(2,1)}.

*O56* When is ρ((a,b),(c,d)) the matrix of an involution? When is it hyperbolic, when elliptic?

*O57* Suppose *r*(*X _{1}*, ...)

Prove that the product

*O58* Suppose we have a pencil of points *l* on which the pair {*P*,*Q*} separates both the pair {*A*,*A'*} and the pair {*B*,*B'*} harmonically.

Prove that {*A'*,*B'*} is a pair of the involution with pairs {*P*,*Q*} and {*A*,*B*}.