Linear Transformation

Apr 2010

I'm trying to find the linear fractional transformation that maps points
z1 = -i, z2 = 0, z3 = i onto points w1 = -1, w2 = i, w3 = 1.

I have set it up using the form:

(w-w1)(w2-w3) / (w-w3)(w2-w1) = (z-z1)(z2-z3) / (z-z3)(z2-z1) such that in my example (w+1)(i-1) / (w-1)(i+1) = (z+i)(0-i) / (z-i)(i).

Multiplying the left side by (i-1) / (i -1), I have i[(w+1) / (w-1)] = the right side of the equation above.

Getting rid of the i on the left side, I have (w+1)/(w-1) = i[(z+i)/(z-i)].

I'm stuck at the point where I have: (w+1)(z-i) = (-iz - 1)(w-1). Is this correct, and what do the final steps look like from here?

Thank you.
Mar 2010
I got
\(\displaystyle \frac{w+1}{w-1}=i \frac{z+i}{z-i}=\frac{zi-1}{z-i}\)
Expressing w
\(\displaystyle w=i \frac{1-z}{1+z}\)