Conformal map
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In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths.
More formally, let [math]\displaystyle{ U }[/math] and [math]\displaystyle{ V }[/math] be open subsets of [math]\displaystyle{ \mathbb{R}^n }[/math]. A function [math]\displaystyle{ f:U\to V }[/math] is called conformal (or anglepreserving) at a point [math]\displaystyle{ u_0\in U }[/math] if it preserves angles between directed curves through [math]\displaystyle{ 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 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 with determinant one). Some authors define conformality to include orientationreversing mappings whose Jacobians can be written as any scalar times any orthogonal matrix.^{[1]}
For mappings in two dimensions, the (orientationpreserving) conformal mappings are precisely the locally invertible complex analytic functions. In three and higher dimensions, 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 or semiRiemannian manifolds.
Conformal maps in two dimensions
If [math]\displaystyle{ U }[/math] is an open subset of the complex plane [math]\displaystyle{ \mathbb{C} }[/math], then a function [math]\displaystyle{ f:U\to\mathbb{C} }[/math] is conformal if and only if it is holomorphic and its derivative is everywhere nonzero on [math]\displaystyle{ U }[/math]. If [math]\displaystyle{ f }[/math] is antiholomorphic (conjugate to a holomorphic function), it preserves angles but reverses their orientation.
In the literature, there is another definition of conformal: a mapping [math]\displaystyle{ f }[/math] which is onetoone and holomorphic on an open set in the plane. The open mapping theorem forces the inverse function (defined on the image of [math]\displaystyle{ 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 onetoone and holomorphic implies having a nonzero derivative. However, the exponential function is a holomorphic function with a nonzero derivative, but is not onetoone since it is periodic.^{[2]}
The Riemann mapping theorem, one of the profound results of complex analysis, states that any nonempty open simply connected proper subset of [math]\displaystyle{ \mathbb{C} }[/math] admits a bijective conformal map to the open unit disk in [math]\displaystyle{ \mathbb{C} }[/math].
Global conformal maps on the Riemann sphere
A map of the Riemann sphere 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, 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.^{[3]} Each is hosted by its own real algebra, ordinary complex numbers, splitcomplex numbers, and dual numbers. The conformal maps are described by linear fractional transformations in each case.^{[4]}
Conformal maps in three or more dimensions
Riemannian geometry
In Riemannian geometry, two Riemannian metrics [math]\displaystyle{ g }[/math] and [math]\displaystyle{ h }[/math] on a smooth manifold [math]\displaystyle{ M }[/math] are called conformally equivalent if [math]\displaystyle{ g = u h }[/math] for some positive function [math]\displaystyle{ u }[/math] on [math]\displaystyle{ M }[/math]. The function [math]\displaystyle{ 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 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 metrics.
Euclidean space
A classical theorem of Joseph Liouville shows that there are much 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 homothety, an isometry, and a special conformal transformation.
Applications
Cartography
In cartography, several named map projections, including the Mercator projection and the stereographic projection are conformal. They are specially useful for use in marine navigation because of its unique property of representing any course of constant bearing as a straight segment. Such a course, known as a rhumb (or, mathematically, a loxodrome) is preferred in marine navigation because ships can sail in a constant compass direction.
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]\displaystyle{ E(z) }[/math], arising from a point charge located near the corner of two conducting planes separated by a certain angle (where [math]\displaystyle{ z }[/math] is the complex coordinate of a point in 2space). 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]\displaystyle{ \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]\displaystyle{ E(w) }[/math], and then mapped back to the original domain by noting that [math]\displaystyle{ w }[/math] was obtained as a function (viz., the composition of [math]\displaystyle{ E }[/math] and [math]\displaystyle{ w }[/math]) of [math]\displaystyle{ z }[/math], whence [math]\displaystyle{ E(w) }[/math] can be viewed as [math]\displaystyle{ E(w(z)) }[/math], which is a function of [math]\displaystyle{ 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 liquid sloshing in tanks.^{[5]}
If a function is harmonic (that is, it satisfies Laplace's equation [math]\displaystyle{ \nabla^2 f=0 }[/math]) over a plane domain (which is twodimensional), 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 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 freesurface flow around a semiinfinite wall, the domain can be mapped to a halfplane in which the solution is onedimensional and straightforward to calculate.^{[6]}
For discrete systems, Noury and Yang presented a way to convert discrete systems root locus into continuous root locus through a wellknow conformal mapping in geometry (aka inversion mapping).^{[7]}
Maxwell's equations
Maxwell's equations are preserved by Lorentz transformations which form a group including circular and hyperbolic rotations. 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 circular angle. The introduction of translations in the Poincare 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: ^{[8]}
 Each fourdimensional solution could be inverted in a fourdimensional hypersphere of pseudoradius [math]\displaystyle{ 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 effect this (that is, replication of all the same trajectories would necessitate departures from geodesic motion because the metric tensor is different). It is often used to try to make models amenable to extension beyond curvature singularities, for example to permit description of the universe even before the Big Bang.
See also
 Biholomorphic map
 Carathéodory's theorem – A conformal map extends continuously to the boundary
 Penrose diagram
 Schwarz–Christoffel mapping – a conformal transformation of the upper halfplane onto the interior of a simple polygon
 Special linear group – transformations that preserve volume (as opposed to angles) and orientation
References
 ↑ Blair, David (20000817). Inversion Theory and Conformal Mapping. The Student Mathematical Library. 9. Providence, Rhode Island: American Mathematical Society. doi:10.1090/stml/009. ISBN 9780821826362.
 ↑ Richard M. Timoney (2004), Riemann mapping theorem from Trinity College Dublin
 ↑ Geometry/Unified Angles at Wikibooks
 ↑ Tsurusaburo Takasu (1941) Gemeinsame Behandlungsweise der elliptischen konformen, hyperbolischen konformen und parabolischen konformen Differentialgeometrie, 2, Proceedings of the Imperial Academy 17(8): 330–8, link from Project Euclid, MR14282
 ↑ Kolaei, Amir; Rakheja, Subhash; Richard, Marc J. (20140106). "Range of applicability of the linear fluid slosh theory for predicting transient lateral slosh and roll stability of tank vehicles". Journal of Sound and Vibration 333 (1): 263–282. doi:10.1016/j.jsv.2013.09.002. Bibcode: 2014JSV...333..263K.
 ↑ Hinton, Edward; Hogg, Andrew; Huppert, Herbert (2020). "Shallow freesurface Stokes flow around a corner". Philosophical Transactions of the Royal Society A 378 (2174). doi:10.1098/rsta.2019.0515. PMID 32507085. Bibcode: 2020RSPTA.37890515H.
 ↑ Noury, Keyvan; Yang, Bingen (2020). "A Pseudo Splane Mapping of Zplane Root Locus". ASME 2020 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers. doi:10.1115/IMECE202023096. ISBN 9780791884546. https://www.researchgate.net/publication/343084262.
 ↑ Warwick, Andrew (2003). Masters of theory : Cambridge and the rise of mathematical physics. University of Chicago Press. pp. 404–424. ISBN 9780226873756. https://archive.org/details/mastersoftheoryc0000warw.
Further reading
 Ahlfors, Lars V. (1973), Conformal invariants: topics in geometric function theory, New York: McGraw–Hill Book Co.
 Constantin Carathéodory (1932) Conformal Representation, Cambridge Tracts in Mathematics and Physics
 Chanson, H. (2009), Applied Hydrodynamics: An Introduction to Ideal and Real Fluid Flows, CRC Press, Taylor & Francis Group, Leiden, The Netherlands, 478 pages, ISBN 9780415492713, http://espace.library.uq.edu.au/view/UQ:191112
 Churchill, Ruel V. (1974), Complex Variables and Applications, New York: McGraw–Hill Book Co., ISBN 9780070108554, https://archive.org/details/complexvariable00chur
 Hazewinkel, Michiel, ed. (2001), "Conformal mapping", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 9781556080104, https://www.encyclopediaofmath.org/index.php?title=C/c024780
 Rudin, Walter (1987), Real and complex analysis (3rd ed.), New York: McGraw–Hill Book Co., ISBN 9780070542341
 Weisstein, Eric W.. "Conformal Mapping". http://mathworld.wolfram.com/ConformalMapping.html.
External links
 Interactive visualizations of many conformal maps
 Conformal Maps by Michael Trott, Wolfram Demonstrations Project.
 Conformal Mapping images of current flow in different geometries without and with magnetic field by Gerhard Brunthaler.
 Conformal Transformation: from Circle to Square.
 Online Conformal Map Grapher.
 Joukowski Transform Interactive WebApp
Original source: https://en.wikipedia.org/wiki/Conformal map.
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