a Mobius transformation
/ MUH-bee-us /
Among all the ways to reshape the complex plane, one small family stands out as the cleanest and most useful — built from nothing fancier than addition, multiplication, and a single division. A Mobius transformation is a map of the form w = (a z + b)/(c z + d), where a, b, c, d are fixed complex numbers chosen so that a*d - b*c is not zero. That last condition just rules out the degenerate case where the formula collapses to a constant. These are the gentlest nontrivial maps of the plane: they are conformal, they are invertible, and they have a beautifully rigid structure.
Read the formula as a recipe. Pick z, multiply the top a z + b, multiply the bottom c z + d, divide. When c = 0 it is just an affine map w = (a/d) z + (b/d) — a rotation, scaling, and translation. When c is not zero the division is what gives Mobius maps their power: a single point (z = -d/c) gets sent off to infinity, and z = infinity comes back to the finite point a/c. To make this seamless we work on the extended plane, the Riemann sphere, where infinity is an ordinary point. On the sphere a Mobius map is a perfect bijection with no exceptions. Its derivative works out to (a*d - b*c)/(c z + d)^2, which is never zero — confirming it is conformal everywhere it is finite.
Mobius transformations are the workhorses of conformal mapping. They send circles-and-lines to circles-and-lines, they are determined by where they send any three points, and they form a group under composition. They give the explicit dictionary between the standard model regions — the upper half-plane, the unit disk, and the plane minus a point. The one thing to keep straight: 'Mobius transformation' (a map of the plane) is a completely different idea from the 'Mobius strip' (a one-sided surface); they share only the name of August Ferdinand Mobius.
The map w = 1/z is a Mobius transformation with a = 0, b = 1, c = 1, d = 0 (so a*d - b*c = -1, nonzero). It sends 0 to infinity, infinity to 0, and 1 to 1; it is its own inverse. Geometrically it is inversion, combined with conjugation undone — and it turns vertical lines into circles through the origin.
w = 1/z is the simplest genuinely non-affine Mobius map; it swaps 0 and infinity.
Two coefficient quadruples give the SAME map if one is a nonzero scalar multiple of the other — (a,b,c,d) and (ta,tb,tc,td) define identical transformations. So the coefficients are only meaningful up to a common factor.