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www.carom-maths.co.uk

Activity 2-15 : The Cross-ratio

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.

.

.

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.

This makes the cross-ratio an invariant,

andof great interest in a field of maths

known as projective geometry.

Projective geometry might be described as

‘the geometry of perspective’.

It is maybe 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.

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

So the cross-ratio is real if and only if

ei(α+β) is, which happens if and only if

α + β = 0 orα + 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!

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