Past exam of the mathematics course of the University of Cambridge / 2016 / ib / Paper 3 / 14F / a

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- Solution a

Solution

Cross-ratio conventions differ by permutations. Use the following ordering, which gives the signs requested in part (c):

r = [z_{1}, z_{2}, z_{3}, z_{4}] = \frac{( z _{1} - z _{4} ) ( z _{2} - z _{3} )}{( z _{1} - z _{2} ) ( z _{4} - z _{3} )} .

Values involving infinity are defined by limits. This is the cross-ratio with second point mapped to zero: the Möbius transformation

T (z) = \frac{( z _{1} - z _{4} ) ( z - z _{2} )}{( z _{1} - z _{2} ) ( z - z _{4} )}

sends

z_{1}, z_{2}, z_{4}

to

1, 0, \infty

and sends

z_{3}

to

r

.

For

M (z) = (a z + b) / (cz + d)

,

a d - b c \neq = 0

, the identity

M (u) - M (v) = \frac{( a d - b c ) ( u - v )}{( c u + d ) ( c v + d )}

shows that all factors cancel in the cross-ratio. This proves Möbius invariance of the cross-ratio, with the cases involving infinity following by limits. Algebraic subtraction also gives

[z_{2}, z_{1}, z_{3}, z_{4}] = 1 - r = \frac{( z _{1} - z _{3} ) ( z _{2} - z _{4} )}{( z _{1} - z _{2} ) ( z _{3} - z _{4} )} .

The latter expression is another common ordering for the cross-ratio; keeping the convention explicit prevents an incorrect reciprocal or sign.

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