PROJECTIVE GEOMETRY COURSE

§ 16: *The main theorem about collineations*

*Proposition :* Let φ be a projective transformation of *P*^{2}. Then there is a pencil of points that are invariant under φ if and only if there is a pencil of lines
that are invariant under φ.

*Proof* : Suppose there is a point *C* such that the lines through *C* are invariant. Let *X '* := φ(*X*). Suppose φ is not the identical mapping.

Make a Desargues configuration with centre *C* and triangles *ABD* and *A ' B ' D ' *.

The intersection points *S*_{1}, *S*_{2}, *S*_{3} of corresponding sides are distinct and according to Desargues collinear.

They are fixed points of φ: for example φ(*S*_{1}) lies on *CS*_{1} because *CS*_{1} is invariant, and φ(*S*_{1}) lies on *A ' D ' *
because *S*_{1} lies on *AD*, so φ(*S*_{1}) = *S*_{1}.

Then the line *a* through *S*_{1}, *S*_{2} and *S*_{3} contains at least three fixed points and hence, according to FT, it is pointwise invariant.

On the other hand, suppose there is a pencil *a* of invariant points.

We distinguish two cases:

i) There is a fixed point *C* outside *a*. In that case, every line *l* through *C* contains two fixed points, namely *C* and the intersection point *l.a*, so *l*
is invariant.

ii) There is no fixed point outside *a*. Take a point *P* outside *a*. Then *P ' * lies outside *a*, too, because φ is 1-1 and *a* is a line of fixed points.

Now let *C* := *a.PP ' *. Then *C* lies on *a*, so *C* is fixed point.

So *PC* = *P ' C* = *P ' C ' *, so *PC* is an invariant line.

Now take *X* outside *a* and outside *PP ' *. Like *PP ' *, *XX ' * is invariant.

So *PP ' *. *XX ' * is fixed point and lies therefore on *a*. Since it also lies on *PP ' *, it is *C*.

So the pencil of lines through *C* is invariant.

*Definition :* We call a projective transformation, not identity, with a line *a* of fixed points and a pencil *C* of invariant lines *central collineation* with *centre*
*C* and *axis* *a*.

*Remark :* For the relation between central collineations and central projections, see §4.

*Problem 63 :* Prove that λ((4,0,-1),(0,3,0),(2,0,1)) is the matrix of a central collineation.

Find the (homogeneous) coordinates of the centre and an equation of the axis.

*Problem 64 :* Find the matrix of the projective transformation φ defined by

λ(0,1,1) → λ(1,0,0), λ(-1,0,1) → λ(1,1,0), λ(0,7,0) → λ(2,2,2), λ(-2,2,3) → λ(1,0,1).

Does φ have fixed points?