1. Activity 2-20: The Cross-ratio
Activity 2-20: The Cross-ratio

2. What happens in the above diagram
if we calculate ?

3. Say A = (p, ap), B = (q, bq), C = (r, cr), D = (s, ds).
So ap = mp + k, bq = mq + k, cr = mr + k, ds = ms +k. .

4. This is the cross-ratio of a, b, c and d.
Strange fact: this answer does not depend on m or k.
So whatever line y = mx + k falls across the four others,
the cross-ratio of lengths will be unchanged.

5. ‘the geometry of perspective’.
This makes the cross-ratio an invariant,
and of great interest in a field of maths
known as projective geometry.
Projective geometry might be described as
‘the geometry of perspective’.
You could argue it is a more fundamental form of geometry
than the Euclidean geometry we generally use.
The cross-ratio has an ancient history;
it was known to Euclid and also to Pappus,
who mentioned its invariant properties.

6. Given four complex numbers z1, z2, z3, z4,
we can define their cross-ratio as .
Theorem: the cross-ratio of four complex numbers is real if and only if the four numbers lie on a straight line or a circle.
Task: certainly 1, i, -1 and –i lie on a circle.
Show the cross-ratio of these numbers is real.

7. (z3-z1) (z2-z4)ei(α+β) = λµ(z3-z4)(z2-z1), or
Proof: we can see that
(z3-z1)eiα = λ(z2-z1),
and (z2-z4)eiβ = µ(z3-z4).
Multiplying these
together gives
(z3-z1) (z2-z4)ei(α+β) = λµ(z3-z4)(z2-z1), or

8. opposite angles in a cyclic quadrilateral,
So the cross-ratio is real if and only if
ei(α+β) is, which happens if and only if
α + β = 0 or α + β = π.
But α + β = 0 implies that α = β = 0,
and z1, z2, z3 and z4 lie on a straight line,
while α + β = π implies that α and β are
opposite angles in a cyclic quadrilateral,
which means that z1, z2, z3 and z4 lie on a circle.
We are done!

9. With thanks to:
Paul Gailiunas
Carom is written by Jonny Griffiths,