Home > Conformal map

w = f(z)

is angle-preserving or (more usually) conformal at a point z₀, if it preserves oriented angles between curves through z₀, as well as their orientation, i.e. direction. Conformal maps preserve both angles and the shapes of infinitesimally small figures.

This is the basic concept for the following applications.

1 Cartography

In cartography, a conformal map projection is a map projection that preserves the angles at all but a finite number of points. Examples include the Mercator projection and the stereographic projection.

2 Complex analysis

An important family of examples comes from complex analysis. If U is an open subset of the complex plane, C, then a function

f : U → C

is conformal if and only if it is holomorphic or antiholomorphic (i.e conjugate to holomorphic), and its derivative is everywhere non-zero on U.

The Riemann mapping theoremThe Riemann mapping theorem in complex analysis states the following: if U is a simply connected open subset of the complex number plane C which is not all of C then there exists a bijective holomorphic conformal map f : U D where D { z in C : z pi;, π, one of the profound results of complex analysis, states that any non-empty open simply connected proper subset of C admits a bijective conformal map to the open unit disk in C.

A map of the extended complex plane (which is conformally equivalent to a sphere) ontoIn mathematics, a surjective function (or onto function or surjection is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. More formally, a function f X &rar itself is conformal if and only if it is a Möbius transformationMobius transformations should not be confused with the Mobius transform. Geometry In mathematics, a Mobius transformation named in honor of August Ferdinand Mobius, is a conformal mapping that is a bijection on the extended complex plane (that is, the com or its conjugate.

3 Riemannian geometry

In Riemannian geometryIn mathematics, Riemannian geometry has at least two meanings, one of which is described in this article and another also called elliptic geometry. In differential geometry, Riemannian geometry is the study of smooth manifolds with Riemannian metrics; i., two Riemannian metrics g and h on smooth manifold M are called conformally equivalent if for g=uh for some positive function u on M. The function u is called conformal factor.

A diffeomorphism between two Riemannian manifolds is called a conformal map if the pulled back metric is conformally equivalent to the original one.

One can also define a conformal structure on a smooth manifold, as a class of conformally equivalent Riemannian metrics.

For example, stereographic projection of a sphere onto the plane augmented with a point at infinity is a conformal map.

Any conformal map from Euclidean space of dimension at least 3 to itself is a composition of a homothety and an isometry.