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Conformal map

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From Wikipedia, the free encyclopedia

Mathematical function that preserves angles

For other uses, seeConformal (disambiguation).

"Conformal projection" redirects here; not to be confused withConformal map projection.

A rectangular grid (top) and its image under a conformal map $f {\displaystyle f}$ (bottom). It is seen that $f {\displaystyle f}$ maps pairs of lines intersecting at 90° to pairs of curves still intersecting at 90°.

A rectangular grid (top) and its image under a conformal map $f {\displaystyle f}$ (bottom). It is seen that $f {\displaystyle f}$ maps pairs of lines intersecting at 90° to pairs of curves still intersecting at 90°.

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Inmathematics, aconformal map is afunction that locally preservesangles, but not necessarily lengths.

More formally, let $U {\displaystyle U}$ and $V {\displaystyle V}$ be open subsets of $\mathbb {R} ^{n}$ . A function $f:U\to V$ is calledconformal (orangle-preserving) at a point $u_{0}\in U$ if it preserves angles between directedcurves through $u_{0}$ , as well as preserving orientation. Conformal maps preserve both angles and the shapes of infinitesimally small figures, but not necessarily their size orcurvature.

The conformal property may be described in terms of theJacobian derivative matrix of acoordinate transformation. The transformation is conformal whenever the Jacobian at each point is a positive scalar times arotation 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.^[1]

For mappings in two dimensions, the (orientation-preserving) conformal mappings are precisely the locally invertiblecomplex 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 betweenRiemannian orsemi-Riemannian manifolds.

In two dimensions

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If $U {\displaystyle U}$ is anopen subset of the complex plane $\mathbb {C}$ , then afunction $f:U\to \mathbb {C}$ is conformalif and only if it isholomorphic and itsderivative is everywhere non-zero on $U {\displaystyle U}$ . If $f {\displaystyle f}$ isantiholomorphic (complex conjugate to a holomorphic function), it preserves angles but reverses their orientation.

In the literature, there is another definition of conformal: a mapping $f {\displaystyle f}$ 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 $f {\displaystyle f}$ ) to be holomorphic. Thus, under this definition, a map is conformalif 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, theinverse function theorem:

(f^{-1}(z_{0}))'={\frac {1}{f'(z_{0})}}

where $z_{0}\in \mathbb {C}$ . However, the exponential function is a holomorphic function with a nonzero derivative, but is not one-to-one since it is periodic.^[2]

TheRiemann mapping theorem, one of the profound results ofcomplex analysis, states that any non-empty opensimply connected proper subset of $\mathbb {C}$ admits abijective conformal map to the openunit disk in $\mathbb {C}$ . Informally, this means that any blob can be transformed into a perfect circle by some conformal map.

Global conformal maps on the Riemann sphere

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A map of theRiemann sphere onto itself is conformal if and only if it is aMöbius 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

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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, ordinarycomplex numbers,split-complex numbers, anddual numbers. The conformal maps are described bylinear fractional transformations in each case.^[4]

In three or more dimensions

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Riemannian geometry

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Euclidean space

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Aclassical theorem ofJoseph Liouville shows that there are far fewer conformal maps in higher dimensions than in two dimensions. Any conformal map from an open subset ofEuclidean space into the same Euclidean space of dimension three or greater can be composed from three types of transformations: ahomothety, anisometry, and aspecial conformal transformation. Forlinear transformations, a conformal map may only be composed ofhomothety andisometry, and is called aconformal linear transformation.

Applications

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Applications of conformal mapping existin aerospace engineering,^[5] in biomedical sciences^[6] (including brain mapping^[7]and genetic mapping^[8]^[9]^[10]),in applied math(for geodesics^[11] and in geometry^[12]),in earth sciences(including geophysics,^[13]geography,^[14]and cartography),^[15]in engineering,^[16]^[17] and in electronics.^[18]

Cartography

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Main article:Conformal map projection

Incartography, several namedmap projections, including theMercator projection and thestereographic projection are conformal. The preservation of compass directions makes them useful in marine navigation.

Physics and engineering

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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, $E(z)$ , arising from a point charge located near the corner of two conducting planes separated by a certain angle (where $z {\displaystyle z}$ is the complex coordinate of a point in 2-space). This problemper 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, $E(w)$ , and then mapped back to the original domain by noting that $w {\displaystyle w}$ was obtained as a function (viz., thecomposition of $E {\displaystyle E}$ and $w {\displaystyle w}$ ) of $z {\displaystyle z}$ , whence $E(w)$ can be viewed as $E(w(z))$ , which is a function of $z {\displaystyle z}$ , 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 theboundary value problem ofliquid sloshing in tanks.^[19]

If a function isharmonic (that is, it satisfiesLaplace's equation $\nabla ^{2}f=0$ ) 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 apotential can be transformed by a conformal map and still remain governed by a potential. Examples inphysics of equations defined by a potential include theelectromagnetic field, thegravitational field, and, influid dynamics,potential flow, which is an approximation to fluid flow assuming constantdensity, zeroviscosity, andirrotational flow. One example of a fluid dynamic application of a conformal map is theJoukowsky 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.^[20]

For discrete systems, Noury and Yang presented a way to convert discrete systemsroot locus into continuousroot locus through a well-known conformal mapping in geometry (akainversion mapping).^[21]

Maxwell's equations

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Maxwell's equations are preserved byLorentz transformations which form a group including circular andhyperbolic rotations. The latter are sometimes called Lorentz boosts to distinguish them from circular rotations. All these transformations are conformal since hyperbolic rotations preservehyperbolic angle, (calledrapidity) and the other rotations preservecircular angle. The introduction of translations in thePoincaré group again preserves angles.

A larger group of conformal maps for relating solutions of Maxwell's equations was identified byEbenezer Cunningham (1908) andHarry Bateman (1910). Their training at Cambridge University had given them facility with themethod of image charges and associated methods of images for spheres and inversion. As recounted by Andrew Warwick (2003)Masters of Theory:^[22]

Each four-dimensional solution could be inverted in a four-dimensional hyper-sphere of pseudo-radius

K {\displaystyle K}

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 inJames Hopwood Jeans textbookMathematical Theory of Electricity and Magnetism.

General relativity

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Ingeneral 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 fromgeodesic motion because themetric tensor is different). It is often used to try to make models amenable to extension beyondcurvature singularities, for example to permit description of the universe even before theBig Bang.