More formally, let and be open subsets of . A function is called conformal (or angle-preserving) at a point if it preserves angles between directed curves through , 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.
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.
Conformal maps in two dimensions
If is an open subset of the complex plane , then a function is conformal if and only if it is holomorphic and its derivative is everywhere non-zero on . If 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 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 ) 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.
The Riemann mapping theorem, one of the profound results of complex analysis, states that any non-empty open simply connected proper subset of admits a bijective conformal map to the open unit disk in .
Global conformal maps on the Riemann sphere
The complex conjugate of a Möbius transformation preserves angles, but reverses the orientation. For example, circle inversions.
Conformal maps in three or more dimensions
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.
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.
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, , arising from a point charge located near the corner of two conducting planes separated by a certain angle (where 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 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, , and then mapped back to the original domain by noting that was obtained as a function (viz., the composition of and ) of , whence can be viewed as , which is a function of , 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.
If a function is harmonic (that is, it satisfies Laplace's equation ) 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.
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: 
- Each four-dimensional solution could be inverted in a four-dimensional hyper-sphere of pseudo-radius 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.
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.
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:
Indeed, any such lies in a particular planar commutative subring, and 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 is one of the three types, slope, hyperbolic, or circular:
- When the subring is isomorphic to the dual number plane, then acts as a shear mapping and preserves the dual angle.
- When the subring is isomorphic to the split-complex number plane, then acts as a squeeze mapping and preserves the hyperbolic angle.
- When the subring is isomorphic to the ordinary complex number plane, then 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.
- 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
- Blair, David (2000-08-17). Inversion Theory and Conformal Mapping. The Student Mathematical Library. 9. Providence, Rhode Island: American Mathematical Society. doi:10.1090/stml/009. ISBN 978-0-8218-2636-2. S2CID 118752074.
- Richard M. Timoney (2004), Riemann mapping theorem from Trinity College, Dublin
- 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. Bibcode:2014JSV...333..263K. doi:10.1016/j.jsv.2013.09.002.
- Noury, Keyvan (2020). "A Psuedo S-plane Mapping of Z-plane Root Locus". Asme Imece2020. doi:10.1115/IMECE2020-23096 (inactive 2020-09-03).
- Warwick, Andrew (2003). Masters of theory : Cambridge and the rise of mathematical physics. University of Chicago Press. pp. 404–424. ISBN 978-0226873756.
- Ahlfors, Lars V. (1973), Conformal invariants: topics in geometric function theory, New York: McGraw–Hill Book Co., MR 0357743
- 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 978-0-415-49271-3
- Churchill, Ruel V. (1974), Complex Variables and Applications, New York: McGraw–Hill Book Co., ISBN 978-0-07-010855-4
- E.P. Dolzhenko (2001) , "Conformal mapping", Encyclopedia of Mathematics, EMS Press
- Rudin, Walter (1987), Real and complex analysis (3rd ed.), New York: McGraw–Hill Book Co., ISBN 978-0-07-054234-1, MR 0924157
- Weisstein, Eric W. "Conformal Mapping". MathWorld.
|Wikimedia Commons has media related to Conformal mapping.|
- 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
- A conformal mapping drawn by M. C. Escher