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[[Image:Conformal map.svg|right|thumb|A rectangular grid (top) and its image under a conformal map ''f'' (bottom). It is seen that ''f'' maps pairs of lines intersecting at 90&deg; to pairs of curves still intersecting at 90&deg;.]]
 
In [[mathematics]], a '''conformal map''' is a [[function (mathematics)|function]] which preserves [[angle]]s. In the most common case the function has a [[Domain of a function|domain]] and [[Range (mathematics)|range]] in the [[complex plane]].
 
More formally, a map,
 
: <math>f: U \rightarrow V\qquad</math> with <math>U,V \subset \mathbb{R}^n</math>
 
is called '''conformal''' (or '''angle-preserving''') at a point <math>u_0</math> if it preserves oriented angles between [[curve]]s through <math>u_0</math> with respect to their [[orientation (mathematics)|orientation]] (i.e., not just the magnitude of the angle). 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 matrix and determinant|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.
 
Conformal maps can be defined between domains in higher dimensional [[Euclidean space]]s, and more generally on a [[Riemannian manifold|Riemannian]] or [[semi-Riemannian manifold]].
 
==Complex analysis==
An important family of examples of conformal maps comes from  [[complex analysis]]. If ''U'' is an [[open set|open subset]] of the complex plane, <math>\scriptstyle\mathbb{C}</math>, then a [[function (mathematics)|function]]
 
: <math>f: U \rightarrow \mathbb{C}</math>
 
is conformal [[if and only if]] it is [[holomorphic function|holomorphic]] and its [[derivative]] is everywhere non-zero on ''U''. If ''f'' is [[antiholomorphic function|antiholomorphic]] (that is, the [[complex conjugate|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 <math>\scriptstyle\mathbb{C}</math> admits a [[bijection|bijective]] conformal map to the open [[unit disk]] in <math>\scriptstyle\mathbb{C}</math>.
 
A map of the [[Riemann sphere|extended complex plane]] (which is [[conformally equivalent]] to a sphere) [[surjection|onto]] itself is conformal if and only if it is a [[Möbius transformation]]. Again, for the [[complex conjugate|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#Circular coordinates|circular coordinates]], keeping the angle the same. See also [[inversive geometry]].
 
==Riemannian geometry==
{{see also|Conformal geometry}}
In [[Riemannian geometry]], two [[Riemannian metric]]s <math>\scriptstyle g</math> and <math>\scriptstyle h</math> on smooth manifold <math>M</math> are called '''conformally equivalent''' if <math>\scriptstyle g = u h </math> for some positive function <math>\scriptstyle u</math> on <math>\scriptstyle M</math>. The function <math>u</math> is called the '''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.  For example, [[stereographic projection]] of a [[sphere]] onto the [[plane (mathematics)|plane]] augmented with a [[point at infinity]] is a conformal map.
 
One can also define a '''conformal structure''' on a smooth manifold, as a class of conformally equivalent [[Riemannian metric]]s.
 
==Higher-dimensional Euclidean space==
A classical theorem of [[Joseph Liouville]] called [[Liouville's theorem (conformal mappings)|Liouville's theorem]] shows the higher-dimensions have less varied conformal maps:
 
Any conformal map on a portion of [[Euclidean space]] of dimension greater than 2 can be composed from three types of transformation: a [[homothetic transformation]], an [[isometry]], and a special conformal transformation. (A ''[[special conformal transformation]]'' is the composition of a reflection and an [[Inversion (geometry)|inversion in a sphere]].)  Thus, the group of conformal transformations in spaces of dimension greater than 2 are much more restricted than the planar case, where the [[Riemann mapping theorem]] provides a large group of conformal transformations.
 
==Uses==
If a function is [[harmonic function|harmonic]] (that is, it satisfies [[Laplace's equation]] <math>\scriptstyle\nabla^2 f=0</math>) over a particular space, and is transformed via a conformal map to another space, the transformation is also harmonic. For this reason, any function which is defined by a [[potential]] can be transformed by a conformal map and still remain governed by a potential.  Examples in [[physics]] of equations defined by a potential include the [[electromagnetic field]], the [[gravitational field]], and, in [[fluid dynamics]], [[potential flow]], which is an approximation to fluid flow assuming constant [[density]], zero [[viscosity]], and [[irrotational vector field|irrotational flow]].  One example of a fluid dynamic application of a conformal map is the [[Joukowsky transform]].
 
Conformal mappings are invaluable for solving problems in engineering and physics that can be expressed in terms of functions of a complex variable but that exhibit inconvenient geometries. By choosing an appropriate mapping, the analyst can transform the inconvenient geometry into a much more convenient one. For example, one may wish to calculate the electric field, <math>\scriptstyle E(z),</math> arising from a point charge located near the corner of two conducting planes separated by a certain angle (where <math>\scriptstyle z</math> is the complex coordinate of a point in 2-space). This problem ''per se'' is quite clumsy to solve in closed form. However, by employing a very simple conformal mapping, the inconvenient angle is mapped to one of precisely pi radians, meaning that the corner of two planes is transformed to a straight line. In this new domain, the problem (that of calculating the electric field impressed by a point charge located near a conducting wall) is quite easy to solve. The solution is obtained in this domain, <math>\scriptstyle E(w),</math> and then mapped back to the original domain by noting that <math>\scriptstyle w</math> was obtained as a function (viz., the [[function composition|composition]] of <math>\scriptstyle E</math> and <math>\scriptstyle w</math>) of <math>\scriptstyle z,</math> whence <math>\scriptstyle E(w)</math> can be viewed as <math>\scriptstyle E(w(z)),</math> which is a function of <math>\scriptstyle z,</math> the original coordinate basis. Note that this application is not a contradiction to the fact that conformal mappings preserve angles, they do so only for points in the interior of their domain, and not at the boundary.
 
A large group of conformal maps for relating solutions of [[Maxwell’s equations]] was identified by [[Ebenezer Cunningham]] (1908) and [[Harry Bateman]] (1910). Their training at Cambridge University had given them facility with the [[method of image charges]] and associated methods of images for spheres and inversion. As recounted by Andrew Warwick (2003) ''Masters of Theory'':
: Each four-dimensional solution could be inverted in a four-dimensional hyper-sphere of pseudo-radius K in order to produce a new solution.
Warwick highlights (pages 404 to 424) this "new theorem of relativity" as a Cambridge response to Einstein, and as founded on exercises using the method of inversion, such as found in [[James Hopwood Jeans]] textbook ''Mathematical Theory of Electricity and Magnetism''.
 
In [[cartography]], several named [[Conformal map projection|map projections]] (including the Mercator projection) are conformal.
 
In [[General Relativity|general relativity]], conformal maps are the simplest and thus most common type of causal transformations. Physically, these describe different universes in which all the same events and interactions are still (causally) possible, but a new additional force is necessary to effect this (that is, replication of all the same trajectories would necessitate departures from [[geodesic]] motion because the [[Metric tensor|metric]] is different). It is often used to try to make models amenable to extension beyond [[Gravitational singularity|curvature singularities]], for example to permit description of the universe even before the [[big bang]].
 
==Alternative angles==
A ''conformal map'' is called that because it preserves the shapes of things (at an infinitesimal scale). The term is based on the Latin prefix ''com-'' (together, with, near) and the Latin noun ''forma'' (shape, appearance).<ref>{{cite web|url=http://www.wordnik.com/words/conformal |title=conformal - definition and meaning |publisher=Wordnik.com |date= |accessdate=2013-09-05}}</ref><ref>{{cite web|url=http://www.myetymology.com/english/conformal.html |title=English etymology of conformal |publisher=myEtymology.com |date= |accessdate=2013-09-05}}</ref><ref>{{cite web|url=http://www.memidex.com/conformal |title=conformal - Memidex dictionary/thesaurus |publisher=Memidex.com |date=2013-06-26 |accessdate=2013-09-05}}</ref> The presumption often is that the shape being preserved is measured by the standard [[angle|Euclidean angle]], say parameterized in degrees or radians. However, in plane mapping there are two other angles to consider: the [[hyperbolic angle]] and the [[slope]], which is the analogue of angle for [[dual number#Geometry|dual numbers]].
 
Suppose <math>\scriptstyle f: U \rightarrow \mathbb{V}</math> is a mapping of surfaces parameterized by <math>\scriptstyle (x,y)\,</math> and <math>\scriptstyle (u,v)\,</math>. The [[Jacobian matrix]] of <math>\scriptstyle f</math> is formed by the four [[partial derivative]]s of <math>\scriptstyle u</math>  and  <math>\scriptstyle v</math>  with respect to <math>\scriptstyle x</math>  and <math>\scriptstyle y</math>.
 
If the Jacobian ''g'' has a non-zero [[determinant]], then <math>f</math> is "conformal in the generalized sense" with respect to one of the three angle types, depending on the [[2 × 2 real matrices|real matrix]] expressed by the Jacobian ''g''.
 
Indeed, any such ''g'' lies in a particular ''planar'' commutative [[subring]], and ''g'' has a polar coordinate form determined by parameters of radial and angular nature. The radial parameter corresponds to a [[similarity (geometry)|similarity mapping]] and can be taken as 1 for purposes of conformal examination. The angular parameter of ''g'' is one of the three types, shear, hyperbolic, or Euclidean:
* When the subring is isomorphic to the [[dual number]] plane, then ''g'' acts as a [[shear mapping]] and preserves the dual angle.
* When the subring is isomorphic to the [[split-complex number]] plane, then ''g'' acts as a [[squeeze mapping]] and preserves the hyperbolic angle.
* When the subring is isomorphic to the ordinary [[complex number]] plane, then ''g'' acts as a [[rotation]] and preserves the Euclidean angle.
While describing analytic [[motor variable#Bireal variable|functions of a bireal variable]], U. Bencivenga and G. Fox have written about conformal maps that preserve the hyperbolic angle.
 
==See also==
* [[Conformal pictures]]
*[[Schwarz–Christoffel mapping]] - a conformal transformation of the upper half-plane onto the interior of a simple polygon.
* [[Penrose diagram]]
* [[Carathéodory's theorem (conformal mapping)|Carathéodory's theorem]] - A conformal map extends continuously to the boundary.
 
== References ==
 
<references />
* {{Citation | last1=Ahlfors | first1=Lars V. | title=Conformal invariants: topics in geometric function theory | publisher=McGraw-Hill Book Co. | location=New York | mr=0357743 | year=1973}}
* {{Citation |last=Chanson |first=H. |authorlink=Hubert Chanson |title=Applied Hydrodynamics: An Introduction to Ideal and Real Fluid Flows |url=http://espace.library.uq.edu.au/view/UQ:191112 |publisher=CRC Press, Taylor & Francis Group, Leiden, The Netherlands, 478 pages |year=2009 |isbn=978-0-415-49271-3}}
* {{springer|id=C/c024780|title=Conformal mapping|author=E.P. Dolzhenko}}
* {{Citation | last1=Rudin | first1=Walter | author1-link=Walter Rudin | title=Real and complex analysis | publisher=McGraw-Hill Book Co. | location=New York | edition=3rd | isbn=978-0-07-054234-1 | mr=924157 | year=1987}}
* {{Citation | last1=Churchill | first1=Ruel V. | title=Complex Variables and Applications | publisher=McGraw-Hill Book Co. | location=New York | isbn=0-07-010855-2 | year=1974}}
* {{MathWorld | urlname=ConformalMapping | title=Conformal Mapping }}
 
== External links ==
* [http://math.fullerton.edu/mathews/c2003/ConformalMappingMod.html Conformal Mapping Module by John H. Mathews]{{Dead link|url=http://math.fullerton.edu/mathews/c2003/ConformalMappingMod.html|date=July 2011}}
* [http://virtualmathmuseum.org/galleryCM.html interactive visualizations of many conformal maps]
* [http://demonstrations.wolfram.com/ConformalMaps/ Conformal Maps] by Michael Trott, [[Wolfram Demonstrations Project]].
* [http://www-m10.ma.tum.de/twiki/bin/view/Lehre/Bsp10_6 Java applet] by Jürgen Richter-Gebert using [[Cinderella (software)|Cinderella]].
* [http://www.math.univ-montp2.fr/SPIP/IMG/jar/ComplexMapStill.jar Java applet] by Christian Mercat to deform pictures; [http://www.math.univ-montp2.fr/SPIP/IMG/jar/ComplexMap.jar MacOSX Java applet] that deforms the video flux from the webcam.
* [http://www.bru.hlphys.jku.at/conf_map/index.html Conformal Mapping images of current flow] in different geometries without and with magnetic field by Gerhard Brunthaler.
* [http://www.flickr.com/photos/sbprzd/362529354 Conformal Transformation: from Circle to Square].
 
[[Category:Riemannian geometry]]
[[Category:Conformal mapping|*]]
[[Category:Cartographic projections]]
[[Category:Angle]]

Latest revision as of 09:16, 8 January 2015

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