PROJECTIVE GEOMETRY COURSE

§ 17: *Classification of the central collineations*

*Proposition :* A central collineation is uniquely determined by its centre *C*, axis *a*, and one pair {*X*,*X ' *} of point and image point, provided
*C*, *X*, *X ' * are collinear and *X* ≠ *X ' *.

*Proof:* Now we can construct for *Y* in *P*^{2} the image point *Y ' * (see *O14*).

*Definition :* We call the central collineation an *elation* if the centre *C* lies on the axis *a*, and otherwise we call it a *homology*.

In the second case we call it a *harmonic homology* if there exists a point *X* such that *X* and *X ' * lie harmonically with respect to *C* and
*S _{X}* , where

As to this, see also the following proposition and its proof.

*Proposition :* A harmonic homology is uniquely determined by its centre and its axis.

*Proof:* Let *X* be a point as in the definition above. Let *Y* be a point not on *CX* and not on *a*. According to *O14* we have the following picture:

So {*Y*,*Y ' *} and {*C*,*S _{Y}*} are also pairs that separate harmonically.

Now by reasoning with

*Proposition :* An elation is uniquely determined by its axis *a* and one pair {*X*,*X ' *} with *X*, ≠ *X ' *.

*Proof:* Then the centre *C* is the intersection point of *a* and *XX ' * (see the proof of the main theorem about collineations).

For the rest, see the first proposition of this section.

*Proposition :* Let φ be an involutory projective transformation. Then φ is a harmonic homology.

*Proof:* φ is involutory, so if φ(*X*) = *X '* , then φ(*X '* ) = *X*.

Now consider the complete quadrangle *AA ' BB ' *.

The lines *AA '* and *BB ' * are invariant, so the intersection point *C* := *AA ' .BB ' * is also invariant.

Since φ interchanges the lines *AB* and *A ' B ' *, *P* := *AB*. *A ' B ' * is invariant.

Likewise, *Q* := *A ' B*. *AB ' * is invariant. So the line *PQ* is invariant, too.

Since *AA ' * and *BB ' * are also invariant, *AA ' .PQ* and *BB ' .PQ* are invariant, too.

So the line *PQ* contains already at least four fixed points, so it is pointwise invariant (propositions of §10).

So φ is a homology with centre *C* and axis *PQ*.

Now {*A*,*A ' *} and {*C*,*S*} both separate harmonically with *S* := *PQ*. *AA ' * (see §12), so φ is a harmonic homology.

*Note:* φ induces a hyperbolice involution on every line through *C*.

*Note:* the inverse proposition also holds.

*Problem 65:* Suppose we have a central collineation with centre *C*, axis *a*, line *x* (not through *C*) and image line *x ' *.

Prove that *x*. *x ' * lies on *a*.

Let *Y* be a point unequal to *C* and not on *a*. Construct *Y ' *.

*Problem 66:* Construct the matrix of the harmonic homology φ with centre λ(0,0,1) and axis x_{3}=0.

*Problem 67:* Give a parameter representation for the matrices of elations with axis λ(1,1,1) and centre λ(0,1,-1).

*Problem 68:* Prove that the following matrix belongs to a central collineation, and determine the centre and the axis: λ((7,-99,-33),(2,-22,-6),(-1,9,-1)).

*Problem 69:* Consider the projective transformation φ with matrix λ((-√2,0,√2),(0,2,0),(√2,0,√2)).

Prove that φ is a harmonic homology and determine the projective coordinates of the centre and the axis.

*Problem 70:* Prove that the matrix of a central collineation φ with axis x_{3} = 0 has the following form:

λ((α,0,β),(0,α,γ),(0,0,1)) with α≠0.

Prove that φ is an elation if α=1 and (β,γ)≠(0,0).

Determine in that case the centre of the elation.