Complex function on circle 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.
Moebius transformations map lines and circles into lines and circles, see

A M&#246;bius 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 $-i$.
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?