If you have a Möbius transformation that is defined by the transformation of 3 points ,by example it maps points 1, −i, −1 to −i, 0, i, respectively, how is it possible to find the exact Möbius map ?
The Möbius map is defined by: \begin{equation}\label{maxwell} f(z) = \frac{az+b}{cz+d} \end{equation}
I am confused because it seems to me like there is only 3 equations for 4 unknowns and so finding a solution without using a parameter seems impossible to me. What am I missing?
There is not a unique solution for $a,b,c,d$; note that any nonzero scalar multiple of $(a,b,c,d)$ gives you the same Möbius transformation (because that scalar cancels in numerator and denominator). This extra degree of freedom in your system of equations explains why three points suffice. So you can simplify your problem somewhat: Unless $f$ maps $0$ to $\infty$, you will have $d\ne 0$ and so you can (rescale and) set $d=1$. Now there will be unique $a,b,c$.
The customary way of solving these problems is to use the fact that Möbius transformations preserve cross-ratio. Denoting the cross ratio of $A,B,C,z$ by $[A:B:C:z]$, if $f$ maps $A$ to $A'$, $B$ to $B'$, $C$ to $C'$, and $z$ to $w$, then we will have $[A':B':C':w]=[A:B:C:z]$, and you can solve for $w=f(z)$ explicitly: $$\frac{\frac{w-A'}{w-B'}}{\frac{C'-A'}{C'-B'}} = \frac{\frac{z-A}{z-B}}{\frac{C-A}{C-B}}.$$