there is a unique mobius transformation taking three points to three points, proof? Watch

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fishpaste
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#1
Report Thread starter 14 years ago
#1
Show that, if z1, z2, z3 and w1, w2, w3 are two triples of
distinct points in C, there exists a unique Mobius transformation that
takes zj to wj [j = 1; 2; 3].

Errr, I can't find my notes on this, and I know there's a really snazzy way to do it. Help would be much appreciated.

Lex!
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fishpaste
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#2
Report Thread starter 14 years ago
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having thought about it myself for a sec, is this correct?

f(z1) = 0
f(z2) = 1
f(z3) = infinity

w1 = g(0)
w2 = g(1)
w3 = g(infinity)
=>
Wj = gf(Zj)

g and f are bijective, therefore so is gf, and so there exists a unique mobius transformation which takes three distinct points to another three distinct points
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