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.
In mathematics, 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.
A diffeomorphism between two Riemannian manifolds is called conformal if the pulled back metric is conformally equivalent to the original one.
The conformal maps preserve angles and shapes of (infinitesimally) small figures. For example, stereographic projection of sphere onto the plane augmented with a point at infinity is a conformal map.
An important family of examples is comming from complex analysis, a function f : U -> C (where U is an open subset of the complex plane C) is conformal if and only if it is holomorphic or anti-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.
Referenced By
Complex geometry | List of mathematical topics | List of mathematical topics (A-C) | List of mathematics topics | Map projection
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