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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.
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.
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 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, mail@jonny.griffiths.net