Conformal map

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In mathematics, a conformal map is a function which preserves angles.

More formally, a map

w = f(z)

is called conformal (or angle-preserving) at 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, but not necessarily their size.

The conformal property may be described in terms of the Jacobian derivative matrix of a coordinate transformation. If the Jacobian matrix of the transformation is everywhere a scalar times a rotation matrix, then the transformation is conformal.

Cartography

In cartography, a conformal map projection is a map projection that preserves the angles at all but a finite number of points. The scale depends on location, but not on direction.

Examples include the Mercator projection and the stereographic projection.

Complex analysis

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

f : UC

is conformal if and only if it is holomorphic and its derivative is everywhere non-zero on U. If f is antiholomorphic (that is, the conjugate to a holomorphic function), it still preserves angles, but it reverses their orientation.

The Riemann mapping theorem, 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) a onto itself is conformal if and only if it is a Möbius transformation. Again, for the conjugate angles are preserved, but orientation is reversed.

An example of the latter is taking the reciprocal of the conjugate, which corresponds to circle inversion with respect to the unit circle. This can also be expressed as taking the reciprocal of the radial coordinate in circular coordinates, keeping the angle the same. See also inversive geometry.

Riemannian geometry

In Riemannian geometry, 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.

Euclidean space

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

Uses

The importance of conformal transformations for electromagnetism was brought to light by Harry Bateman in 1910.

See also

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