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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.

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  • $\begingroup$ Under the assumption that any given circle is mapped onto another circle under this transformation, here is what you can do: Pick three points on the given circle. Find their images using $f(z)$. Now you have three points of the new circle. That information is enough to find the equation of the new circle, it's just algebra $\endgroup$
    – imranfat
    Commented Jun 6, 2018 at 19:47

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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 $-i$.

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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?

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  • $\begingroup$ No, it is not correct! $\endgroup$
    – mikekh123
    Commented Jun 9, 2018 at 11:07

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