# Proof that there is a unique linear fractional transformation that maps three distinct points to three distinct points in the extended complex plane.

The following is a theorem and a proof from Complex Variables by Herb Silverman. The bolded are parts that I don't understand in the proof.

Theorem: Given three distinct points, $z_1,z_2, z_3$ in the extended $z$ plane and three distinct points $w_1,w_2,w_3$ in the extended $w$ plane, there exists a unique bilinear transformation $w=t(z)$ such that $T(z_k)=w_k$ for $k=1,2,3$.

Proof. We first assume that none of the six points is $\infty$. Let $w=T(z)=\frac{az+b}{cz+d}$. We wish to solve for $a,b,c,$ and $d$ in terms of $z_1,z_2,z_3,w_1,w_2,$ and $w_3$. This sounds more complicated than it is. For $k=1,2,3,$ we have $$w-w_k=\frac{az+b}{cz+d}-\frac{az_k+b}{cz_k+d}=\frac{(ad-bc)(z-z_k)}{(cz+d)(cz_k+d)}(3.9)$$

From (3.9) we obtain $$\frac{w-w_1}{w-w_3}=\frac{cz_1+d}{cz_3+d}\frac{z_2-z_3}{z_2-z_1} (3.10)$$

Replacing $z$ by $z_2$ and $w$ by $w_2$ in (3.10) leads to $$\frac{w_2-w_3}{w_2-w_1}=\frac{cz_1+d}{cz_3+d}\frac{z_2-z_3}{z_2-z_1} (3.11)$$

Multiplying (3.10) by (3.11) we have $$\frac{(w-w_1)(w_2-w_3)}{(w-w_3)(w_2-w_1)}=\frac{(z-z_1)(z_2-z_3)}{(z-z_3)(z_2-z_1)} (3.12)$$ Solving for $w$ in terms of $z$ and the six points gives the desired transformation. If one of the points were the point at $\infty$, say $z_3=\infty$, (3.12) would be modified by taking the limit as $z_3$ approached $\infty$. In this case, we would have $$\frac{(w-w_1)(w_2-w_3)}{(w-w_3)(w_2-w_1)}=\frac{z-z_1}{z_2-z_1} (3.13)$$

Corollary. Given three distinct points, $z_1,z_2,z_3$ in the extended $z$ plane there exists a unique bilinear transformation $w=T(z)$ such that $T(z_1)=0, T(z_2)=1, T(z_3)=\infty$ and it is given by $$w=\frac{(z-z_1)(z_2-z_3)}{(z-z_3)(z_2-z_1)}$$.

Firstly, I don't see how to obtain the desired transformation. I did find an expression for $w$ in terms of all other variables (which looks complicated) and found $T(z_1)=w_1$ and $T(z_2)=w_2$. However, I don't get $T(z_3)=w_3$. In fact, if we look at (3.12) and plug in $z_3$ in place of $z$, the fraction on the right hand side isn't even defined since the denominator is $0$...So I don't see how we can get the desired result from this expression. Actually, I don't understand why the author sets out to find an expression for $w$ by multiplying fractions. What may be the reasoning behind this?

Moreover, I don't understand the part when one of the points is $\infty$. It makes sense to take the limit as $z_3 \to \infty$, but how does this bring the expression (3.13)?

Finally, for the corollary, again this is similar to the first question, I think we're supposed to use (3.12) and simply plug in the appropriate $w_k$ values but in case of $T(z_3)=\infty$, the expression just doesn't make sense to me. How does $\frac{(w-w_1)(w_2-\infty)}{(w-\infty)(w_2-w_1)}$ make sense?

I would greatly appreciate it if anyone clarifies the above questions to me, I'm having trouble reading this page because of these.

I have not read all your post, but perhaps a nicer way of proving the result is to start with the corollary: first prove that there is a unique linear fractional transformations which maps 3 distinct points to (0,1,$\infty$). It is not difficult to write down the expression (which you have), and for unicity you can use e.g. the fact that any linear fractional transformation which fixes more than 2 points is the identity.
Now recall that all linear fractional transformations are invertible, therefore you also know that you can map (0,1,$\infty$) to any triple of distinct points.