PROJECTIVE GEOMETRY COURSE

§ 9: *Separation and cross ratio*

Given four distinct points *A*,*B*,*C*,*D* on a line *l*.

After choosing an orientation of *l*, the distances have a sign such that *AB* = - *BA*, etc.

We define the *cross ratio* of *A*,*B*,*C*,*D* (in this order) by (*A*,*B*;*C*,*D*) = (*AC*/*AD*)/(*BC*/*BD*).
The cross ratio is independent of the orientation of *l*.

We say the pairs {*A*,*B*} and {*C*,*D*} *separate* each other if (*A*,*B*;*C*,*D*) is negative (this means that one of the points *C*,*D*
is lying between *A* and *B*, and the other is not. See *O36*.

*Proposition : *Separation and cross ratio are invariant under a perspectivity from a line onto a line.

*Proof* : (see *O35*, too)

That separation is invariant, is evident (but distinguish between the case that *P* is lying between *l* and *l'* and the other case). Furthermore:

Let *H* be the foot point of the line through *P* perpendicular to *l*. If we forget the signs, we have:

(*AC*/*AD*)/(*BC*/*BD*) = ((*AC* . *HP*/2)/(*AD* . *HP*/2))/((*BC* . *HP*/2)/(*BD* . *HP*/2)) =
((area Δ*APC*)/(area Δ*APD*))/((area Δ*BPC*)/(area Δ*BPD*)) =

((*AP.CP* sin∠*APC*)/(*AP.DP* sin∠*APD*))/((*BP.CP* sin∠*BPC*)/(*BP.DP* sin∠*BPD*)) =
((sin∠*APC*)/(sin∠*APD*))/((sin∠*BPC*)/(sin∠*BPD*)) =

((sin∠*A'P'C'*)/(sin∠*A'P'D'*))/((sin∠*B'P'C'*)/(sin∠*B'P'D'*)) = (*A'C'*/*A'D'*)/(*B'C'*/*B'D'*).

For four lines *a*,*b*,*c*,*d* through a point *P* we define the cross ratio (*a*,*b*;*c*,*d*) as
(*a.l* , *b.l* ; *c.l* , *d.l*), where *l* is an arbitrary line, but not going through *P*.
The pairs {a,b} and {c,d} separate each other if {a.l , b.l} and {c.l , d.l} separate each other.

Now you yourself can define the cross ratio of four points on the line at infinity.

The following proposition is evident;
*Proposition* : separation and cross ratio are invariant under projectivities.

*O35* Prove the last proposition but one in the case that *P* and/or *D* are points at infinity, and in the case that *l* is the line at infinity.

*O36* Let (*X*_{1},*X*_{2};*X*_{3},*X*_{4}) = λ. Calculate
(*X*_{σ(1)},*X*_{σ(2)};*X*_{σ(3)},*X*_{σ(4)}) for each permutation σ of {1,2,3,4}.

How many distinct values do we get? (Hint: (*X*_{1},*X*_{2};*X*_{3},*X*_{4}) +
(*X*_{4},*X*_{2};*X*_{3},*X*_{1}) = 1.)

*O37* Let *X*_{1},*X*_{2},*X*_{3},*X*_{4} be four points on a line *l*, and
*x*_{1},*x*_{2},*x*_{3},*x*_{4} four lines through a point *L*, whilst *X*_{i} and *x*_{i} have the same projective
coordinates. Prove that (*X*_{1},*X*_{2};*X*_{3},*X*_{4}) =
(*x*_{1},*x*_{2};*x*_{3},*x*_{4}).

*O38* Define the function f from a line *l* into the set of real numbers (plus ∞ (=-∞)) by f(*X*) = (*X*,*B*;*C*,*D*), where *B*,*C*,*D*
are three distinct fixed points on *l*.

Prove that f is a continuous bijection with values as in the picture below: