# Conformal map

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In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths.

## Contents

More formally, let ${\displaystyle U}$ and ${\displaystyle V}$ be open subsets of ${\displaystyle \mathbb {R} ^{n}}$. A function ${\displaystyle f:U\to V}$ is called conformal (or angle-preserving) at a point ${\displaystyle u_{0}\in U}$ if it preserves angles between directed curves through ${\displaystyle u_{0}}$, 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 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 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 semi-Riemannian manifolds.

## Conformal maps in two dimensions

If ${\displaystyle U}$ is an open subset of the complex plane ${\displaystyle \mathbb {C} }$, then a function ${\displaystyle f:U\to \mathbb {C} }$ is conformal if and only if it is holomorphic and its derivative is everywhere non-zero on ${\displaystyle U}$. If ${\displaystyle f}$ 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 ${\displaystyle f}$ 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 ${\displaystyle f}$) 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. However, the exponential function is a holomorphic function with a nonzero derivative, but is not one-to-one since it is periodic. [2]

The Riemann mapping theorem, one of the profound results of complex analysis, states that any non-empty open simply connected proper subset of ${\displaystyle \mathbb {C} }$ admits a bijective conformal map to the open unit disk in ${\displaystyle \mathbb {C} }$.

### 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.

## Conformal maps in three or more dimensions

### Riemannian geometry

In Riemannian geometry, two Riemannian metrics ${\displaystyle g}$ and ${\displaystyle h}$ on a smooth manifold ${\displaystyle M}$ are called conformally equivalent if ${\displaystyle g=uh}$ for some positive function ${\displaystyle u}$ on ${\displaystyle M}$. The function ${\displaystyle u}$ 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 on a portion of 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. These enjoy the property that the distortion of shapes can be made as small as desired by making the diameter of the mapped region small enough.

### 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, ${\displaystyle E(z)}$, arising from a point charge located near the corner of two conducting planes separated by a certain angle (where ${\displaystyle z}$ 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 ${\displaystyle \pi }$ 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, ${\displaystyle E(w)}$, and then mapped back to the original domain by noting that ${\displaystyle w}$ was obtained as a function (viz., the composition of ${\displaystyle E}$ and ${\displaystyle w}$) of ${\displaystyle z}$, whence ${\displaystyle E(w)}$ can be viewed as ${\displaystyle E(w(z))}$, which is a function of ${\displaystyle z}$, 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 the boundary value problem of liquid sloshing in tanks. [3]

If a function is harmonic (that is, it satisfies Laplace's equation ${\displaystyle \nabla ^{2}f=0}$) 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 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.

### Maxwell's equations

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: [4]

Each four-dimensional solution could be inverted in a four-dimensional hyper-sphere of pseudo-radius ${\displaystyle K}$ 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 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.

## Pseudo-Riemannian geometry

In differential geometry a mapping is conformal when angles are preserved. When the angle is related to the metric, it is sufficient for the mapping to result in a metric that is proportional to the original, as expressed above for Riemannian geometry or in the case of a conformal manifold with the type of metric tensor used in general relativity. An elementary consideration of surface mapping and linear algebra reveals potentially three types of angles: circular angle, hyperbolic angle, and slope:

Suppose ${\displaystyle f:U\rightarrow \mathbb {C} }$ is a mapping of surfaces parameterized by ${\displaystyle (x,y)}$ and ${\displaystyle (u,v)}$. The Jacobian matrix of ${\displaystyle f}$ is formed by the four partial derivatives of ${\displaystyle u}$ and ${\displaystyle v}$ with respect to ${\displaystyle x}$ and ${\displaystyle y}$.

If the Jacobian ${\displaystyle g}$ has a non-zero determinant, then ${\displaystyle f}$ is conformal with respect to one of the three angle types, depending on the real matrix expressed by the Jacobian ${\displaystyle g}$.

Indeed, any such ${\displaystyle g}$ lies in a particular planar commutative subring, and ${\displaystyle g}$ has a polar decomposition determined by parameters of radial and angular nature. The radial parameter corresponds to a similarity mapping and can be taken as 1 for purposes of conformal examination. The angular parameter of ${\displaystyle g}$ is one of the three types, slope, hyperbolic, or circular:

• When the subring is isomorphic to the dual number plane, then ${\displaystyle g}$ acts as a shear mapping and preserves the dual angle.
• When the subring is isomorphic to the split-complex number plane, then ${\displaystyle g}$ acts as a squeeze mapping and preserves the hyperbolic angle.
• When the subring is isomorphic to the ordinary complex number plane, then ${\displaystyle g}$ acts as a rotation and preserves the circular angle.

While describing analytic functions of a bireal variable, U. Bencivenga and G. Fox have written about conformal maps that preserve the hyperbolic angle. In general, a linear fractional transformation on any one of the types of complex plane listed provides a conformal map.

## Related Research Articles

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including the branches of hydrodynamics, thermodynamics, and particularly quantum mechanics. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering.

In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are smooth.

In complex analysis, the Riemann mapping theorem states that if U is a non-empty simply connected open subset of the complex number plane C which is not all of C, then there exists a biholomorphic mapping f from U onto the open unit disk

In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together.

In mathematics, the open unit disk around P, is the set of points whose distance from P is less than 1:

In mathematics, an isometry is a distance-preserving transformation between metric spaces, usually assumed to be bijective.

In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature Kp) depends on a two-dimensional plane σp in the tangent space at a point p of the manifold. It is the Gaussian curvature of the surface which has the plane σp as a tangent plane at p, obtained from geodesics which start at p in the directions of σp. The sectional curvature is a smooth real-valued function on the 2-Grassmannian bundle over the manifold.

In mathematics, specifically differential calculus, the inverse function theorem gives a sufficient condition for a function to be invertible in a neighborhood of a point in its domain: namely, that its derivative is continuous and non-zero at the point. The theorem also gives a formula for the derivative of the inverse function. In multivariable calculus, this theorem can be generalized to any continuously differentiable, vector-valued function whose Jacobian determinant is nonzero at a point in its domain, giving a formula for the Jacobian matrix of the inverse. There are also versions of the inverse function theorem for complex holomorphic functions, for differentiable maps between manifolds, for differentiable functions between Banach spaces, and so forth.

In mathematics, conformal geometry is the study of the set of angle-preserving (conformal) transformations on a space.

The theory of functions of several complex variables is the branch of mathematics dealing with complex valued functions

This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology.

In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic.

In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions from the open unit disk to itself. The lemma is less celebrated than stronger theorems, such as the Riemann mapping theorem, which it helps to prove. It is, however, one of the simplest results capturing the rigidity of holomorphic functions.

Affine geometry, broadly speaking, is the study of the geometrical properties of lines, planes, and their higher dimensional analogs, in which a notion of "parallel" is retained, but no metrical notions of distance or angle are. Affine spaces differ from linear spaces in that they do not have a distinguished choice of origin. So, in the words of Marcel Berger, "An affine space is nothing more than a vector space whose origin we try to forget about, by adding translations to the linear maps." Accordingly, a complex affine space, that is an affine space over the complex numbers, is like a complex vector space, but without a distinguished point to serve as the origin.

Geometric function theory is the study of geometric properties of analytic functions. A fundamental result in the theory is the Riemann mapping theorem.

In the mathematical theory of functions of one or more complex variables, and also in complex algebraic geometry, a biholomorphism or biholomorphic function is a bijective holomorphic function whose inverse is also holomorphic.

In mathematics, the Abel–Jacobi map is a construction of algebraic geometry which relates an algebraic curve to its Jacobian variety. In Riemannian geometry, it is a more general construction mapping a manifold to its Jacobi torus. The name derives from the theorem of Abel and Jacobi that two effective divisors are linearly equivalent if and only if they are indistinguishable under the Abel–Jacobi map.

In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives: extrinsically, relating to their embedding in Euclidean space and intrinsically, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss, who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space.

In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane, the complex plane plus a point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers plus a value ∞ for infinity. With the Riemann model, the point "∞" is near to very large numbers, just as the point "0" is near to very small numbers.

In mathematics, a harmonic morphism is a (smooth) map between Riemannian manifolds that pulls back real-valued harmonic functions on the codomain to harmonic functions on the domain. Harmonic morphisms form a special class of harmonic maps i.e. those that are horizontally (weakly) conformal.

## References

1. Blair, David (2000-08-17). Inversion Theory and Conformal Mapping. The Student Mathematical Library. Providence, Rhode Island: American Mathematical Society. ISBN   978-0-8218-2636-2.
2. Richard M. Timoney (2004), Riemann mapping theorem from Trinity College, Dublin
3. Kolaei, Amir; Rakheja, Subhash; Richard, Marc J. (2014-01-06). "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.
4. Warwick, Andrew (2003). . University of Chicago Press. pp.  404–424. ISBN   978-0226873756.