Circle Inversion with programming languages

Question

How can I find Circle Inversion of any region with programming tools?

For an example; consider problem of finding circle inversion for below region :

Any ideas or solution would be appreciated, thanks.

Answer 1

Inversions are easily described using complex numbers. We shall therefore assimilate points with complex numbers and represent hyperbolic points by complex numbers with modulus strictly less than one. The inversion with respect to a circle with center c and radius r is the function which transforms any complex z into a complex z' defined by :

Answer 2

I guess you won't be needing this answer any more, since it is no longer involved in the best answer on this question of yours. But in any case, here is what you do to perform inversion in the unit circle: given coordinates x1, y1 you compute the coordinates of the inverted point as

sq = x1*x1 + y1*y1
x2 = x1/sq
y2 = y1/sq

If the circle you want to use for inversion is not the unit circle, you can apply one transformation up front and its inverse afterwards. So if you have a circle with center mx, my and radius r you can do

x2 = (x1 - mx)/r
y2 = (y1 - my)/r
sq = x2*x2 + y2*y2
x3 = x2/sq
y3 = y2/sq
x4 = x3*r + mx
y4 = y3*r + my