# Thread: Find Möbius transformation

1. ## Find Möbius transformation

Find the Möbius transformation that maps the points $\displaystyle z=1,2,3$ to $\displaystyle w=i,-i,-1.$What are its fixed points?

By a Theorem, I know there's a transformation so that $\displaystyle T(z_j)=w_j$ whenever $\displaystyle z_1,z_2,z_3,w_1,w_2,w_3\in\mathbb C-\{0,1,\infty\},$ so I took a generic transformation given by $\displaystyle w(z)=\frac{az+b}{cz+d},$ so I did $\displaystyle i=\frac{a+b}{c+d},$ $\displaystyle -i=\frac{2a+b}{2c+d}$ and $\displaystyle -1=\frac{3a+b}{3c+d},$ then I think I need to find a relation, or something, how to solve this?

2. ## Re: Find Möbius transformation

Originally Posted by Homing
$\displaystyle i=\frac{a+b}{c+d},$ $\displaystyle -i=\frac{2a+b}{2c+d}$ and $\displaystyle -1=\frac{3a+b}{3c+d},$
All right. Now write the corresponding linear system on the unknowns $\displaystyle a,b,c,d$ , you'll find solutions of the form $\displaystyle a=\alpha d,\;b=\beta d,\;c=\gamma d$ . So the transformation is $\displaystyle w=\frac{\alpha d z+\beta d}{\gamma dz+d}=\frac{\alpha z+\beta }{\gamma z +1}$ .

3. ## Re: Find Möbius transformation

Okay so that includes that those constants could have the imaginary unit right?

4. ## Re: Find Möbius transformation

Originally Posted by Homing
Okay so that includes that those constants could have the imaginary unit right?
Right. Also, after finding $\displaystyle \alpha,\beta,\gamma$ you can easily check if the solution is correct, $\displaystyle w(1)=\ldots=i$ etc.