conformal geometry

Imagine a map of the world that distorts areas wildly — Greenland looks as big as Africa — yet at every single point the compass directions are still correct, so that wherever two roads cross at a right angle on the ground they cross at a right angle on the map. Such a map preserves angles while abandoning size. Conformal geometry is the geometry of exactly this property: it studies what is preserved by the angle-preserving maps, the transformations that may stretch and bend a figure freely but always keep the angle between two crossing curves intact.

In the Erlangen scheme, conformal geometry — in its tidiest, globally consistent form on the plane — is the geometry whose group is the Mobius group: the transformations of the plane-plus-one-extra-point that are built by composing rotations, translations, uniform scalings, and one new move, inversion in a circle. Every such map preserves angles, including the size and sense of the angle at which any two curves meet, but it need not preserve length, area, straightness, or even the distinction between lines and circles. The key invariants are angles and, more subtly, the cross-ratio of four points and the family of 'lines-and-circles' taken together; the things lost include length, area, and the separate notions of line versus circle, since inversion can turn a line into a circle. This is why these maps are studied on the inversive plane, the ordinary plane with a single point at infinity added, where lines are just circles that happen to pass through that point.

Conformal geometry is the meeting place of geometry and complex analysis: in the plane the orientation-preserving conformal maps are exactly the maps given by holomorphic functions with nonzero derivative, so the whole theory of analytic functions is conformal geometry in disguise. It is indispensable wherever angles matter but size does not — fluid flow, electrostatics, the design of angle-true navigational charts, and the standard models of hyperbolic geometry, whose isometries are Mobius transformations. A frequent confusion is to expect conformal maps to look 'similar' to the original; they preserve angles only infinitesimally, so a large square can be bent into a curvy region that is locally angle-true everywhere yet globally nothing like a square.