Activity 2-20: The Cross-ratio www.carom-maths.co.uk Activity 2-20: The Cross-ratio

What happens in the above diagram if we calculate ?

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. .

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.

‘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.

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.

(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

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!

With thanks to: Paul Gailiunas Carom is written by Jonny Griffiths, hello@jonny.griffiths.net