Conformal map Guide, Meaning , Facts, Information and Description
In mathematics, a mapping
- w = f(z)
This is the basic concept for the following applications.
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2 Complex analysis 3 Riemannian geometry 4 See also |
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.
An important family of examples comes from complex analysis. If U is an open subset of the complex plane, C, then a function
Complex analysis
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 theorem, one of the profound results of complex analysis, states that any non-empty open simply connected proper subset of C admits a conformal map onto the open unit disk in C.
A map of the extended complex plane (which is conformally equivalent to a sphere) onto itself is conformal if and only if it is a Möbius transformation or its conjugate.
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.
Any conformal map from Euclidean space of dimension at least 3 to itself is a composition of a homothety and an isometry.