Activity 2-20: The Cross-ratio

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www.carom-maths.co.uk
Activity 2-20 The Cross-ratio
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What happens in the above diagram if we
calculate ?
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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.
.
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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.
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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.
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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.
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Proof we can see that (z3-z1)eia ?(z2-z1),
and (z2-z4)eiß µ(z3-z4). Multiplying these
together gives
(z3-z1) (z2-z4)ei(aß) ?µ(z3-z4)(z2-z1), or
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So the cross-ratio is real if and only if
ei(aß) is, which happens if and only if
a ß 0 or a ß p.
But a ß 0 implies that a ß 0, and z1,
z2, z3 and z4 lie on a straight line, while a
ß p implies that a and ß 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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With thanks to
Paul Gailiunas
Carom is written by Jonny Griffiths,
mail_at_jonny.griffiths.net