View source for Conformal map

{{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°.]]
{{Complex analysis sidebar}}

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.

==In two dimensions==
If <math>U</math> is an [[open set|open subset]] of the complex plane <math>\mathbb{C}</math>, then a [[function (mathematics)|function]] <math>f:U\to\mathbb{C}</math> is conformal [[if and only if]] it is [[holomorphic function|holomorphic]] and its [[derivative]] is everywhere non-zero on <math>U</math>. If <math>f</math> is [[antiholomorphic function|antiholomorphic]] ([[complex conjugate|complex conjugate]] to a holomorphic function), it preserves angles but reverses their orientation.

In the literature, there is another definition of conformal: a mapping <math>f</math> which is one-to-one and holomorphic on an open set in the plane. The open mapping theorem forces the inverse function (defined on the image of <math>f</math>) to be holomorphic. Thus, under this definition, a map is conformal [[if and only if]] it is biholomorphic. The two definitions for conformal maps are not equivalent. Being one-to-one and holomorphic implies having a non-zero derivative. In fact, we have the following relation, the [[inverse function theorem]]:
::<math>(f^{-1}(z_0))'=\frac{1}{f'(z_0)}</math>
where <math>z_0 \in \mathbb{C}</math>. However, the exponential function is a holomorphic function with a nonzero derivative, but is not one-to-one since it is periodic.<ref>Richard M. Timoney (2004), [https://www.maths.tcd.ie/~richardt/414/414-ch7.pdf Riemann mapping theorem] from [[Trinity College Dublin]]</ref>

The [[Riemann mapping theorem]], one of the profound results of [[complex analysis]], states that any non-empty open [[simply connected]] proper subset of <math>\mathbb{C}</math> admits a [[bijection|bijective]] conformal map to the open [[unit disk]] in <math>\mathbb{C}</math>. Informally, this means that any blob can be transformed into a perfect circle by some conformal map.

=== Global conformal maps on the Riemann sphere ===

A map of the [[Riemann sphere]] [[surjection|onto]] itself is conformal if and only if it is a [[Möbius transformation]].

The complex conjugate of a Möbius transformation preserves angles, but reverses the orientation. For example, [[Inversive geometry#Circle inversion|circle inversions]].

===Conformality with respect to three types of angles===
In plane geometry there are three types of angles that may be preserved in a conformal map.<ref>{{wikibooks-inline|Geometry/Unified Angles}}</ref> Each is hosted by its own real algebra, ordinary [[complex number]]s, [[split-complex number]]s, and [[dual number]]s. The conformal maps are described by [[linear fractional transformation#Conformal property|linear fractional transformations]] in each case.<ref>Tsurusaburo Takasu (1941) [https://projecteuclid.org/euclid.pja/1195578674 Gemeinsame Behandlungsweise der elliptischen konformen, hyperbolischen konformen und parabolischen konformen Differentialgeometrie, 2], [[Japan Academy|Proceedings of the Imperial Academy]] 17(8): 330–8, link from [[Project Euclid]], {{mr|id=14282}}</ref>

==In three or more dimensions==

===Riemannian geometry===
{{see also|Conformal geometry}}
In [[Riemannian geometry]], two [[Riemannian metric]]s <math>g</math> and <math>h</math> on a smooth manifold <math>M</math> are called '''conformally equivalent''' if <math> g = u h </math> for some positive function <math>u</math> on <math>M</math>. The function <math>\sqrt 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.

===Euclidean space===
A [[Liouville's theorem (conformal mappings)|classical theorem]] of [[Joseph Liouville]] shows that there are far fewer conformal maps in higher dimensions than in two dimensions. Any conformal map from an open subset of [[Euclidean space]] into the same Euclidean space of dimension three or greater can be composed from three types of transformations: a [[homothetic transformation|homothety]], an [[isometry]], and a [[special conformal transformation]]. For [[Linear map|linear transformations]], a conformal map may only be composed of [[homothetic transformation|homothety]] and [[isometry]], and is called a [[conformal linear transformation]].

==Applications==

Applications of conformal mapping exist
in aerospace engineering,<ref>{{Cite journal |last1=Selig |first1=Michael S. |last2=Maughmer |first2=Mark D. |date=1992-05-01 |title=Multipoint inverse airfoil design method based on conformal mapping |url=https://arc.aiaa.org/doi/10.2514/3.11046 |journal=AIAA Journal |volume=30 |issue=5 |pages=1162–1170 |doi=10.2514/3.11046 |bibcode=1992AIAAJ..30.1162S |issn=0001-1452|url-access=subscription }}</ref> 
in biomedical sciences<ref>{{Cite journal |last1=Cortijo |first1=Vanessa |last2=Alonso |first2=Elena R. |last3=Mata |first3=Santiago |last4=Alonso |first4=José L. |date=2018-01-18 |title=Conformational Map of Phenolic Acids |journal=The Journal of Physical Chemistry A |volume=122 |issue=2 |pages=646–651 |doi=10.1021/acs.jpca.7b08882 |issn=1520-5215 |pmid=29215883|bibcode=2018JPCA..122..646C }}</ref> 
(including brain mapping<ref>{{Cite web |title=Properties of Conformal Mapping |url=https://www.researchgate.net/figure/Properties-of-Conformal-Mapping-Conformal-mappings-transform-infinitesimal-circles-to_fig1_228640184}}</ref>
and genetic mapping<ref>{{Cite web |title=7.1 GENETIC MAPS COME IN VARIOUS FORMS |url=https://www.informatics.jax.org/silver/chapters/7-1.shtml |access-date=2022-08-22 |website=www.informatics.jax.org}}</ref><ref>{{Cite journal |url=https://hcvalidate.perfdrive.com/?ssa=71fed169-69c9-41b8-9877-416719f0d113&ssb=80077267131&ssc=https%3A%2F%2Fiopscience.iop.org%2Farticle%2F10.1088%2F1478-3975%2F13%2F5%2F05LT01%2Fpdf&ssi=e1a50ce6-8427-48c7-9e70-ece2b2f11df9&ssk=support@shieldsquare.com&ssm=73836051457188970102906276668717&ssn=bd1de37a5005eecad6da1d8bb5de43cdae96922a58cb-972c-4c61-bbf00b&sso=ce98748a-7da0b84595a0eb14c80eef4551a4a5763a991c17cd7fabdb&ssp=79762773411661113911166113142686120&ssq=67286667226024947028072260265646682776545&ssr=MjA4LjgwLjE1My4yNA==&sst=ZoteroTranslationServer/WMF%20(mailto:noc@wikimedia.org)&ssv=&ssw=&ssx=W10= |access-date=2022-08-22 |journal=Physical Biology |doi=10.1088/1478-3975/13/5/05lt01| title=Leaf growth is conformal | year=2016 | last1=Alim | first1=Karen | last2=Armon | first2=Shahaf | last3=Shraiman | first3=Boris I. | last4=Boudaoud | first4=Arezki | volume=13 | issue=5 | article-number=05LT01 | pmid=27597439 | arxiv=1611.07032 | bibcode=2016PhBio..13eLT01A | s2cid=9351765 }}</ref><ref>{{Cite journal |last1=González-Matesanz |first1=F. J. |last2=Malpica |first2=J. A. |date=2006-11-01 |title=Quasi-conformal mapping with genetic algorithms applied to coordinate transformations |url=https://www.sciencedirect.com/science/article/pii/S0098300406000161 |journal=Computers & Geosciences |language=en |volume=32 |issue=9 |pages=1432–1441 |doi=10.1016/j.cageo.2006.01.002 |bibcode=2006CG.....32.1432G |issn=0098-3004|url-access=subscription }}</ref>),
in applied math
(for geodesics<ref>{{Cite journal |last1=Berezovski |first1=Volodymyr |last2=Cherevko |first2=Yevhen |last3=Rýparová |first3=Lenka |date=August 2019 |title=Conformal and Geodesic Mappings onto Some Special Spaces |journal=Mathematics |language=en |volume=7 |issue=8 |page=664 |doi=10.3390/math7080664 |issn=2227-7390|doi-access=free |hdl=11012/188984 |hdl-access=free }}</ref> 
and in geometry<ref>{{Cite journal |last=Gronwall |first=T. H. |date=June 1920 |title=Conformal Mapping of a Family of Real Conics on Another |journal=Proceedings of the National Academy of Sciences |language=en |volume=6 |issue=6 |pages=312–315 |doi=10.1073/pnas.6.6.312 |issn=0027-8424 |pmc=1084530 |pmid=16576504|bibcode=1920PNAS....6..312G |doi-access=free }}</ref>),
in earth sciences
(including geophysics,<ref>{{Cite web |title=Mapping in a sentence (esp. good sentence like quote, proverb...) |url=https://sentencedict.com/mapping_10.html |access-date=2022-08-22 |website=sentencedict.com}}</ref>
geography,<ref>{{Cite web |title=EAP - Proceedings of the Estonian Academy of Sciences – Publications. |url=https://kirj.ee/proceedings-of-the-estonian-academy-of-sciences-publications/ |access-date=2022-08-22 |language=en-GB}}</ref>
and cartography),<ref>{{Cite journal |last=López-Vázquez |first=Carlos |date=2012-01-01 |title=Positional Accuracy Improvement Using Empirical Analytical Functions |journal=Cartography and Geographic Information Science |volume=39 |issue=3 |pages=133–139 |doi=10.1559/15230406393133 |bibcode=2012CGISc..39..133L |s2cid=123894885 |issn=1523-0406}}</ref>
in engineering,<ref>{{Cite journal |last1=Calixto |first1=Wesley Pacheco |last2=Alvarenga |first2=Bernardo |last3=da Mota |first3=Jesus Carlos |last4=Brito |first4=Leonardo da Cunha |last5=Wu |first5=Marcel |last6=Alves |first6=Aylton José |last7=Neto |first7=Luciano Martins |last8=Antunes |first8=Carlos F. R. Lemos |date=2011-02-15 |title=Electromagnetic Problems Solving by Conformal Mapping: A Mathematical Operator for Optimization |journal=Mathematical Problems in Engineering |language=en |volume=2010 |article-number=e742039 |doi=10.1155/2010/742039 |issn=1024-123X|doi-access=free |hdl=10316/110197 |hdl-access=free }}</ref><ref>{{Cite journal |last=Leonhardt |first=Ulf |date=2006-06-23 |title=Optical Conformal Mapping |journal=Science |language=en |volume=312 |issue=5781 |pages=1777–1780 |doi=10.1126/science.1126493 |pmid=16728596 |bibcode=2006Sci...312.1777L |s2cid=8334444 |issn=0036-8075|doi-access=free }}</ref> 
and in electronics.<ref>{{Cite journal |last1=Singh |first1=Arun K. |last2=Auton |first2=Gregory |last3=Hill |first3=Ernie |last4=Song |first4=Aimin |date=2018-07-01 |title=Estimation of intrinsic and extrinsic capacitances of graphene self-switching diode using conformal mapping technique |url=https://ui.adsabs.harvard.edu/abs/2018TDM.....5c5023S |journal=2D Materials |volume=5 |issue=3 |page=035023 |doi=10.1088/2053-1583/aac133 |bibcode=2018TDM.....5c5023S |s2cid=117531045 |issn=2053-1583}}</ref>

===Cartography===
{{main|Conformal map projection}}
In [[cartography]], several named [[map projection]]s, including the [[Mercator projection]] and the [[stereographic projection]] are conformal. The preservation of compass directions makes them useful in marine navigation.

===Physics and engineering===
Conformal mappings are invaluable for solving problems in engineering and physics that can be expressed in terms of functions of a complex variable yet 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>E(z)</math>, arising from a point charge located near the corner of two conducting planes separated by a certain angle (where <math>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 <math>\pi</math> 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>E(w)</math>, and then mapped back to the original domain by noting that <math>w</math> was obtained as a function (''viz''., the [[function composition|composition]] of <math>E</math> and <math>w</math>) of <math>z</math>, whence <math>E(w)</math> can be viewed as <math>E(w(z))</math>, which is a function of <math>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. Another example is the application of conformal mapping technique for solving the [[boundary value problem]] of [[Slosh dynamics|liquid sloshing]] in tanks.<ref>{{Cite journal|last1=Kolaei|first1=Amir|last2=Rakheja|first2=Subhash|last3=Richard|first3=Marc J.|date=2014-01-06|title=Range of applicability of the linear fluid slosh theory for predicting transient lateral slosh and roll stability of tank vehicles|journal=Journal of Sound and Vibration|volume=333|issue=1|pages=263–282|doi=10.1016/j.jsv.2013.09.002|bibcode=2014JSV...333..263K}}</ref>

If a function is [[harmonic function|harmonic]] (that is, it satisfies [[Laplace's equation]] <math>\nabla^2 f=0</math>) over a plane domain (which is two-dimensional), and is transformed via a conformal map to another plane domain, 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]] that can be used to examine the field of flow around a Joukowsky airfoil.

Conformal maps are also valuable in solving nonlinear partial differential equations in some specific geometries. Such analytic solutions provide a useful check on the accuracy of numerical simulations of the governing equation. For example, in the case of very viscous free-surface flow around a semi-infinite wall, the domain can be mapped to a half-plane in which the solution is one-dimensional and straightforward to calculate.<ref>{{cite journal |first1=Edward |last1=Hinton |first2=Andrew |last2=Hogg |first3=Herbert |last3=Huppert |year=2020 | title=Shallow free-surface Stokes flow around a corner | journal=Philosophical Transactions of the Royal Society A | volume=378 |issue=2174 |doi=10.1098/rsta.2019.0515|pmid=32507085 |pmc=7287310|bibcode=2020RSPTA.37890515H }}</ref>

For discrete systems, Noury and Yang presented a way to convert discrete systems [[root locus]] into continuous [[root locus]] through a well-known conformal mapping in geometry (aka [[Inversive geometry|inversion mapping]]).<ref>{{cite book |first1=Keyvan |last1=Noury |first2=Bingen |last2=Yang |year=2020 |chapter=A Pseudo S-plane Mapping of Z-plane Root Locus |chapter-url=https://www.researchgate.net/publication/343084262 |title=ASME 2020 International Mechanical Engineering Congress and Exposition |publisher=American Society of Mechanical Engineers |doi=10.1115/IMECE2020-23096|isbn=978-0-7918-8454-6 |s2cid=234582511 }}</ref>

===Maxwell's equations===
[[Maxwell's equations]] are preserved by [[Lorentz transformation]]s which form a group including circular and [[hyperbolic rotation]]s. The latter are  sometimes called Lorentz boosts to distinguish them from circular rotations. All these transformations are conformal since hyperbolic rotations preserve [[hyperbolic angle]], (called [[rapidity]]) and the other rotations preserve [[angle|circular angle]]. The introduction of translations in the [[Poincaré group]] again preserves angles. 

A larger 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'':
<ref>{{cite book|last1=Warwick|first1=Andrew|title=Masters of theory: Cambridge and the rise of mathematical physics|url=https://archive.org/details/mastersoftheoryc0000warw|url-access=registration|date=2003|publisher=[[University of Chicago Press]]|pages=[https://archive.org/details/mastersoftheoryc0000warw/page/404 404–424]|isbn=978-0-226-87375-6}}</ref>

: Each four-dimensional solution could be inverted in a four-dimensional hyper-sphere of pseudo-radius <math>K</math> in order to produce a new solution.
Warwick highlights 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''.

===General relativity===
In [[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 affect this (that is, replication of all the same trajectories would necessitate departures from [[geodesic]] motion because the [[metric tensor (general relativity)|metric tensor]] 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]].

==See also==
* [[Biholomorphic map]]
* [[Carathéodory's theorem (conformal mapping)|Carathéodory's theorem]] – A conformal map extends continuously to the boundary
* [[Penrose diagram]]
* [[Schwarz–Christoffel mapping]] – a conformal transformation of the upper half-plane onto the interior of a simple polygon
* [[Special linear group]] – transformations that preserve volume (as opposed to angles) and orientation

== References ==
{{reflist}}

== Further reading ==
* {{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}}
* [[Constantin Carathéodory]] (1932) ''Conformal Representation'', Cambridge Tracts in Mathematics and Physics
* {{Citation |last=Chanson |first=H. |author-link=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}}
* {{Citation | last1=Churchill | first1=Ruel V. | author-link=Ruel Churchill | title=Complex Variables and Applications | publisher=McGraw–Hill Book Co. | location=New York | isbn=978-0-07-010855-4 | year=1974 | url-access=registration | url=https://archive.org/details/complexvariable00chur }}
* {{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}}
* {{MathWorld | urlname=ConformalMapping | title=Conformal Mapping }}

== External links ==
{{Commons category|Conformal mapping}}
* [https://virtualmathmuseum.org/ConformalMaps/index.html Interactive visualizations of many conformal maps]
* [http://demonstrations.wolfram.com/ConformalMaps/ Conformal Maps] by Michael Trott, [[Wolfram Demonstrations Project]].
* [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.
* [https://www.flickr.com/photos/sbprzd/362529354 Conformal Transformation: from Circle to Square].
* [http://www.davidbau.com/conformal Online Conformal Map Grapher].
* [http://airfoil.dimanov.com/ Joukowski Transform Interactive WebApp]

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[[Category:Conformal mappings| ]]
[[Category:Riemannian geometry]]
[[Category:Map projections]]
[[Category:Angle]]