Editing Conformal map (section)

{{short description|Mathematical function that preserves angles}}
{{other uses|Conformal (disambiguation)}}
{{redirect-distinguish|Conformal projection|Conformal map projection}}
[[Image:Conformal map.svg|right|thumb|A rectangular grid (top) and its image under a conformal map <math>f</math> (bottom). It is seen that <math>f</math> maps pairs of lines intersecting at 90° to pairs of curves still intersecting at 90°.]]
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In [[mathematics]], a '''conformal map''' is a [[function (mathematics)|function]] that locally preserves [[angle]]s, but not necessarily lengths.

More formally, let <math>U</math> and <math>V</math> be open subsets of <math>\mathbb{R}^n</math>. A function <math>f:U\to V</math> is called '''conformal''' (or '''angle-preserving''') at a point <math>u_0\in U</math> if it preserves angles between directed [[curve]]s through <math>u_0</math>, as well as preserving orientation. Conformal maps preserve both angles and the shapes of infinitesimally small figures, but not necessarily their size or [[curvature]].

The conformal property may be described in terms of the [[Jacobian matrix and determinant|Jacobian]] derivative matrix of a [[coordinate transformation]]. The transformation is conformal whenever the Jacobian at each point is a positive scalar times a [[rotation matrix]] ([[Orthogonal matrix|orthogonal]] with determinant one). Some authors define conformality to include orientation-reversing mappings  whose Jacobians can be written as any scalar times any orthogonal matrix.<ref>{{Cite book|title=Inversion Theory and Conformal Mapping|volume = 9|last=Blair|first=David|s2cid = 118752074|date=2000-08-17|publisher=American Mathematical Society|isbn=978-0-8218-2636-2|series=The Student Mathematical Library|location=Providence, Rhode Island|doi = 10.1090/stml/009}}</ref>

For mappings in two dimensions, the (orientation-preserving) conformal mappings are precisely the locally invertible [[Holomorphic function|complex analytic]] functions. In three and higher dimensions, [[Liouville's theorem (conformal mappings)|Liouville's theorem]] sharply limits the conformal mappings to a few types.

The notion of conformality generalizes in a natural way to maps between [[Riemannian manifold|Riemannian]] or [[semi-Riemannian manifold]]s.