I would like to understand how to choose the right Mobius transformation. For example, the Möbius transformation that maps upper half-plane onto the unit disk is:

z --> (z-i)/(z+i)

The Möbius transformation that maps the unit disk one to one onto the right half-plane is:

z--> (1+z)/(1-z)

I would like to understand how one gets to this Möbius transformations. Thank you!