Conformal map

In cartography, a map projection is called conformal if it preserves the angles at all but a finite number of points. Examples include the Mercator projection and the stereographic projection. It is impossible for a map projection to be both conformal and equal-area.

More generally, in mathematics, for any manifold with a conformal structure (which assigns an angle to intersections of differentiable curves), a conformal mapping is any homeomorphism which preserves the conformal structure. For example, the cartographic example of projecting a 2-sphere onto the plane augmented with a point at infinity is a conformal map.

In particular, in complex analysis, a conformal map is a function f : U -> C (where U is an open subset of the complex numbers C) which maintains angles, and therefore the shape of small figures. A function f is conformal if and only if it is holomorphic and its derivative is everywhere non-zero.

An important statement about conformal maps is the Riemann mapping theorem.

A map of the extended complex plane onto itself (the word onto means surjective) is conformal iff it is a Möbius transformation.