Mobius transformation of the unit circle and the unit disc

Question

Find what happens to the unit circle and the unit disc under the Mobius transformation w = $\frac {z+i} {z-i}$

I believe this is the Cayley map and that it simply rotates the circle and disk, but I'm not sure. Can anyone help?

Answer 1

it is another circle. i forget which one so i test three points.

$-i \to 0$ and $1 \to i$ and $i \to \infty$. the image is a vertical line through the origin

for another approach i recommend the Projective Geometry textbook of Veblen