a Mobius transformation
/ MUH-bee-us /
A Mobius transformation is the simplest kind of map that mixes the ordinary moves of the plane with the one genuinely new conformal move, inversion. Think of it as the most general way to rearrange the plane while keeping every crossing angle correct and never breaking the family of lines-and-circles apart. Viewed through complex numbers it has a strikingly compact formula, which is why it is also called a linear fractional transformation.
Precisely, treating the plane as the complex numbers and adding a single point at infinity, a Mobius transformation is a map of the form z -> (a z + b) / (c z + d), where a, b, c, d are complex constants with a d minus b c not equal to zero (that condition keeps the map invertible and non-degenerate). The point where the denominator vanishes is sent to infinity, and infinity is sent to a over c, so the formula behaves cleanly everywhere on the inversive plane. Every Mobius transformation is conformal — it preserves the angle at which any two curves meet — and it carries the whole family of lines and circles to itself: a line or circle always maps to a line or circle, though a line can become a circle and vice versa. It also preserves the cross-ratio of any four points, the fundamental Mobius invariant. Remarkably, these maps are exactly the compositions of translations, rotations, scalings, and inversions, so they are the building blocks of conformal geometry. A clean fact for computation: a Mobius transformation is completely determined by where it sends any three distinct points, and there is always exactly one sending three given points to three other given points.
Mobius transformations are central across mathematics. They form the symmetry group of the inversive plane and the Riemann sphere; they are the isometries of the standard models of hyperbolic geometry, so the rigid motions of the hyperbolic plane are literally these fractional formulas; and they appear throughout complex analysis. One caveat to guard against: although a Mobius transformation maps the set of lines-and-circles to itself, it does not generally preserve centres or radii, and it usually does not send a straight line to a straight line — straightness is not a Mobius invariant, only membership in the combined line-or-circle family is.
The map z -> 1 / z is a Mobius transformation, with a = 0, b = 1, c = 1, d = 0 (so a d minus b c equals minus 1, nonzero). It sends 0 to infinity and infinity to 0, fixes the points 1 and minus 1, and turns the imaginary axis (a line) into itself but turns many other lines into circles. It is conformal, so it preserves every crossing angle.
z -> (a z + b)/(c z + d): conformal, maps lines-and-circles to lines-and-circles, preserves cross-ratio.
A Mobius transformation maps the combined family of lines and circles to itself, but it need not send a line to a line — straightness is not preserved; only membership in the line-or-circle family is.