to carry a straight line in z-plane onto the unit circle $|w|=1$
I could restrict this transformation to Moebius transformation which satisfies $$ w=\frac{\alpha\, z+\beta}{\gamma\, z+ \delta}\qquad for \,\,every \,\, \alpha,\,\beta,\,\gamma,\,\delta \in \mathbb C. $$ $$ where \qquad \alpha\gamma-\beta\delta \neq 0 \,\,,\,\,\gamma \ne 0, \,and \quad |\alpha|=|\gamma| $$ Are there any conditions to add?