How to determine a map which sends $(1,-1,0)\to(\infty,0,i)$?

Question

I know that the uninque bilinear map sending $(z_1,z_2,z_3)\to (\infty,0,1)$ is given by $$T(z)=\frac{(z-z_2)(z_3-z_1)}{(z-z_1)(z_3-z_2)}.$$ Well, could any one tell me how to determine a map which sends $(1,-1,0)\to(\infty,0,i)$?

Answer 1

You know how to find maps to $(\infty, 0, 1)$. Have you thought how that might that be used in your quest?

Answer 2

Got one theorem saying The Unique Bi-Linear Map $w=f(z)$ which sends $w_1,w_2,w_3\to z_1,z_2,z_3$ is given by $$\frac{(w-w_2)(w_3-w_1)}{(w-w_1)(w_3-w_2)}=\frac{(z-z_2)(z_3-z_1)}{(z-z_1)(z_3-z_2)}$$