PROJECTIVE GEOMETRY COURSE

§ 13: *Three kinds of projectivities from a line onto itself*

*Remark :* If φ is a projectivity from a pencil of points *l* onto itself, and φ has three fixed points, then φ is identity.
This follows from the fundamental theorem.

Likewise: a projectivity from a pencil of lines *L* onto itself with three invariant lines is identitity.

*Definition :* Let φ be a projectivity from a pencil of points *l* onto itself (respectively from a pencil of lines *L* onto itself).

i) If φ has no fixed points (invariant lines), then we say φ is *elliptic*.

ii) If φ has exactly one fixed point (invariant line), we say φ is *parabolic*.

i) If φ has two fixed points (invariant lines), we say φ is *hyperbolic*.

There exist projectivities of each kind:

i) Let φ be the projectivity *l* → *l* with, for three given points *A*, *B*, *C* on *l*, φ(*A*) = *B*, φ(*B*) = *C*,
φ(*C*) = *A*.

Supoose there is a fixed point *D*. Then (*A*, *B*; *C*, *D*) = (*B*, *C*; *A*, *D*), so d = 1 - 1/d (vgl *O36*). Since for all real
numbers d, and also for d=∞, we have d ≠ 1 - 1/d, we find a contradiction.

So φ is elliptic.

ii) (see the picture below)

Let λ be the perspectivity from *l* onto *m* with center *R*, and μ the perspectivity from *m* onto *l* with center *S*.

Then μoλ is a parabolic projectivity from *l* onto *l* (*P* is the only fixed point).

Make the dual construction, too.

iii) Let *A*, *B*, *C*, *D* be distinct points on *l*. Let φ be the projectivity *l* → *l* with φ(*A*) = *A*,
φ(*B*) = *B*, φ(*C*) = *D*. Dan is φ niet de identieke afbeelding, maar φ heeft wel twee verschillende dekpunten. Dus is φ hyperbolisch.

Using the third proposition of §10, show that the projectivities from a line *l* onto itself correspond to the regular 2 by 2 matrices.

How can we determine for a given matrix A what the kind of the corresponding projectivity is?

Do the parabolic projectivities, plus identity, form a subgroup of the group of projectivities from *l* onto *l*?

*O52* Suppose we have a hyperbolic projectivity π with π(*A*) = *A*, π(*B*) = *B'*, π(*C*) = *C'*.
Construct the second fixed point.

*O53* Let π be a parabolic projectivity with fixed point *A*, and let π(*B*) = *B'*. Construct the image point of *X*.

*O54* Given a line *l*: λ(1,2,1) with on it the points *A*: λ(2,-1,0), *B*: λ(0,1,-2), and *C*: λ(-1,1,-1).

Let φ be the projectivity from *l* onto *l* that maps *A* to itself, and *B* and *C* in each other.

Give the matrix of φ on the base * b*,