www.carom-maths.co.uk. Activity 2-15 : 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 . . .
Activity 2-15 : 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.
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).
(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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