Editing Conformal map (section)

==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}}</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 |url=https://pubmed.ncbi.nlm.nih.gov/29215883/ |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 | pages=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}}</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 |pages=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 |url=https://doi.org/10.1559/15230406393133 |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 |pages=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 |pages=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-know 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=978-0226873756}}</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]].