the circle-preserving property

Draw any circle, or any straight line, on the plane, and run it through a Mobius transformation. What comes out is again a circle or a straight line. This is the single most memorable thing Mobius maps do: they preserve the family 'lines-and-circles' as a whole. A circle might become a different circle, or it might straighten out into a line; a line might curl up into a circle. But you will never get an ellipse, a parabola, a figure-eight, or anything fancier. The class of all lines and circles is sealed shut under these maps.

The clean way to see this is to treat a line as 'a circle that happens to pass through the point at infinity'. On the Riemann sphere, lines and circles really are the same kind of object — both are circles on the sphere (a line is the image of a circle through the north pole under stereographic projection). Then the statement becomes simply: a Mobius transformation maps circles on the sphere to circles on the sphere. The proof factors the map into building blocks — translation, rotation, scaling, and the inversion w = 1/z — and checks each one. Translations, rotations, and scalings obviously keep lines and circles. The only one needing work is inversion, and a short calculation (the equation of a general line-or-circle is A(x^2 + y^2) + B x + C y + D = 0, and 1/z swaps the roles of A and D) shows inversion keeps the family too. Compose the pieces and the whole map inherits the property.

This is why Mobius maps are the right tool for transplanting problems between disks, half-planes, and slit regions: their boundaries are made of circular arcs and line segments, and those are exactly what survives. A practical caveat worth stating: 'circle goes to circle' does NOT mean 'center goes to center'. Mobius maps badly distort which point is the center, and they do not preserve radii or distances; only the shape-class 'line-or-circle' is preserved, not the metric data inside it.