A Möbius map w = (az+b)/(cz+d) sends circles and lines to circles and lines. Draw points, circles, or lines in the left plane and watch their images appear on the right.
A modern JavaScript + Canvas port of a 1999 Java applet by Terence Tao
(originally MobiusApp.java / Mobius.java).
Tip: the unit circle (centre 0, radius 1) under inversion maps to itself; a line not through 0 maps to a circle through the image of ∞.