The Bilinear map that sends $(z_1,z_2,z_3)\to (\infty,0,1)$ is given by $f(z)=\frac{(z-z_2)(z_3-z_1)}{(z-z_1)(z_3-z_2)}$

Can I extract any expression which sends the other around? I mean

$(\infty,0,1)\to (z_1,z_2,z_3) $ ?

I know one expression sending any $(z_1,z_2,z_3)\to (w_1,w_2,w_3)$ but I can not remember that expression. is there any way to construct meself from the above one which I always remember by $(23132)$

  • $\begingroup$ Just compute the inverse of the map you have supplied. $\endgroup$ – mrf Sep 12 '13 at 10:42
  • $\begingroup$ How to do that @mrf $\endgroup$ – Marso Sep 13 '13 at 5:50

You make the Ansatz $$f(z)={az+b \over cz+d}$$Then $f(\infty)=\frac ac=z_1, f(0)=\frac bd=z_2$ and $f(1)=\frac {a+b}{c+d}=z_3$. Now solve for $a,b,c,d$ noting that the answer is unique up to linear multiples (i.e. if $(a,b,c,d)$ a solution, then also $(\lambda a,\lambda b,\lambda c,\lambda d)$ is a solution for $\lambda\neq 0$).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.