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November 3rd, 2015, 02:43 PM   #1
Joined: Nov 2015
From: UK

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How do you show a Möbius transformation maps circles to circles and lines?

So I have been trying to prove this for a week now. It is driving me crazy

So the general equation of a circle is : (x-a)^2 + (y-b)^2= r^2
and you put that in the form
zz ̅ – (z+z ̅) α + (z-z ̅ )αi = r^2- |α|^2
which in the w-plane is
1/(ww ̅ ) - ((w ̅+w)/(ww ̅ ))α + ( (w ̅-w)/(ww ̅ )) αi = r^2- |α|^2

now if the centre α = o - how do you prove that it maps to a line?

Last edited by skipjack; November 3rd, 2015 at 03:16 PM.
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