Complex function on circle

Question

I have $f(z)= -\frac{z-i}{z+i}$ which, I believe, will map a circle into another circle. So if I start with a circle in the complex plane with centre $(x,y)$ of radius $r$ how do I determine the centre and radius after applying $f(z)$?

I can see that I substitute $x + iy$ for $z$ in the function $f(z)$ but then I have just an expression rather than an equation - don't I? (My university level maths is over $40$ years ago...). Thanks.

Answer 1

Moebius transformations map lines and circles into lines and circles, see

A Möbius transformation maps circles and lines to circles and lines. What exactly does that mean?

Remark for a reference: Your map is (the negative of) the Cayley transform $$ z\mapsto \frac{z-i}{z+i}, $$ which maps the upper half plane biholomorphically to the open unit disk. So $f(z)$ maps $\infty$ to $-1$, $1$ to $i$ and $-1$ to $-i$.

Answer 2

If I apply f to the center I get the new center, f(x, y) -> c'

If I apply f to a point on the circle I get a point on the new circle. e.g.

f(x - r, y) -> p

and so, from c' and p, can determine the radius.

Isn't this correct?