We introduce a quantity which is invariant under the projective group P(2).
This quantity is very significant geometrically, and admits several interesting
interpretations. The study of this invariant goes back to Greek geometry,
but it looks much more natural in a projective setting.
Suppose that A is a ppoint. This is represented by a line through the origin
in R^{3}. If a is a direction vector for this line,
then, as an element
of RP^{2}, A = [a] . Of course, we could replace
a by a', where a' = λa for any nonzero
real number λ.
Also, a pline L is represented by a plane Π through the origin in R^{3}. A ppoint
A = [a] lies on L if and only if a lies on Π. If B = [b] and C = [c]
also lie on L,
i.e. A,B,C are collinear ppoints, then a, b and c lie on Π.
Provided that A ≠ B, {a, b} is a basis for Π, so that there
exist unique real numbers α and β with
c = αa + βb.
Further, if C ≠ A,B, then
α and β will be nonzero. Of course, α, β, and even their ratio, will depend upon
the choice of the vectors
a, b and c on their respective lines in R^{3}.
If we introduce a fourth ppoint on L, then we do have an invariant.


The crossratio porism
Suppose that A,B,C,D are distinct collinear ppoints, and that a,b,c,d are chosen
so that A=[a], B = [b], C = [c] and D = [d]. Then we have
(1) There exist unique nonzero real numbers α,β,γ and δ such that
c = αa + βb, and d = γa + δb.
(2) The ratio βγ/αδ does not depend on the choice of a,b,c,d.
Of course, part (1) follows easily from the earlier remarks about collinear ppoints.
proof of part(2)
Since the final ratio depends only upon the ppoints and not on the choice of
vectors, we can make the
Definition
If A,B,C,D are distinct, collinear ppoints then the crossratio (A,B,C,D) is the
value of the ratio described in the crossratio porism.
Remarks
(1) The value of the crossratio will depend on the order of the points.
In the notation of the porism, in calculating (B,A,C,D), we have
c = βb + αa , and d = δb + γa,
so that (B,A,C,D) = αδ/βγ = 1/(A,B,C,D).
The reader may care to verify that we also have (A,B,D,C) = 1/(A,B,C,D).
(2) Since we require that A,B,C,D are distinct, α,β,γ,δ are nonzero.
Thus (A,B,C,D) ≠ 0.
(3) Since C and D are distinct ppoints, (α,β) ≠ λ(γ,δ) for any λ
Thus (A,B,C,D) ≠ 1.
(4) A crossratio can take any value other than 0 and 1.
To see this, choose c = a+b, and d = λa+b, where λ ≠ 0,1.
The conditions on λ ensure that the ppoints A,B,C,D are distinct.
Then (A,B,C,D) = 1.λ/1.1 = λ.
Suppose that t is a projective transformation. Then t maps plines to plines.
Thus, if A,B,C,D are collinear ppoints, then t(A),t(B),t(C),t(D) are also collinear,
so their crossratio is defined. We shall write t(A,B,C,D) for (t(A),t(B),t(C),t(D)).
The projective crossratio theorem
Crossratio is invariant under the projective group P(2).
proof
It is often useful to note that, if A,B,C lie on a pline L, then there is a unique
ppoint D on L with a given value of (A,B,C,D). The uniqueness is guaranteed by
Theorem PI1
If A,B,C,D,E are collinear ppoints such that (A,B,C,D) = (A,B,C,E), then D = E.
proof of theorem PI1
To use this invariant to obtain euclidean results, we need to consider
embeddings of RP^{2}.

