carom-2-20.ppt maths. Project management

Maths project

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

2. What happens in the above diagram
if we calculate ?

3. 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.
.

4. Strange fact: this answer does not depend on mor k.
So whatever line y = mx + k falls across the four others,
the cross-ratio of lengths will be unchanged.
. This is the cross-ratio
of a, b, c and d.

5. 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’.
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 Euclidand also to Pappus,
who mentioned itsinvariantproperties.

6. 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.
Given four complex numbers z
1, z
2, z
3, z
4,
we can define their cross-ratioas
Task: certainly 1, i, -1 and –i lie on a circle.
Show the cross-ratio of these numbers is real.
.

7. Proof: we can see that
(z
3-z
1)e
iα
= λ(z
2-z
1),
and (z
2-z
4)e
iβ
= µ(z
3-z
4).
Multiplying these
together gives
(z
3-z
1) (z
2-z
4)e
i(α+β)
= λµ(z
3-z
4)(z
2-z
1), or

8. But α + β = 0 implies that α = β = 0,
and z
1, z
2, z
3and z
4lie on a straight line,
while α + β = π implies that α and β are
opposite angles in a cyclic quadrilateral,
which means that z
1, z
2, z
3and z
4lie on a circle.
So the cross-ratio is real if and only if
e
i(α+β)
is, which happens if and only if
α + β = 0 orα + β = π.
We are done!

9. With thanks to:
Paul Gailiunas
Carom is written by Jonny Griffiths, [email protected]