Lines that don't pass through origin, map$(f(z)=\frac{1}{z})$ to disks. How do I map the line$(1,t); t\in (-\infty, \infty)$ to a disk?

Question

Lines that don't pass through origin, map$(f(z)=\frac{1}{z})$ to disks. How do I map the line$(1,t); t\in (-\infty, \infty)$ to a disk?

How about the general case $(t,at+b)$, $t\in(-\infty, \infty)$?

In class I have mapped disks to other disks and disks that pass through the origin to lines, but not this, and was wondering how this is done.. I would like a familiar equation of a disk like $|z-a|=r$ or even better maybe $(x-a_1)+(y-a_2)=r^2$

Answer 1

All you have to do is map $\infty$ to a finite point and make sure that no pint on the line is mapped to $\infty$. For a vertical line $\operatorname{Re}(z)=r$ you can take $$ f(z)=\frac{a\,z+b}{c\,z+d},\quad c\ne0,\quad c(r+i\,t)+d\ne0\quad\forall t\in\mathbb{R}. $$ I hope this is enough for you to figure out the general case.