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 (i.e. mapping a tangent basis to a basis of the same 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. If the Jacobian matrix of the transformation is everywhere a scalar times an orientation-preserving rotation matrix, then the transformation is conformal.
An important family of examples of conformal maps comes from complex analysis. 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 (that is, the conjugate to a holomorphic function), it still preserves angles, but it reverses their orientation.
In the literature, there is another definition of conformal maps; a map defined on an open set is said to be conformal if it is one-to-one and holomorphic. Since a one-to-one map defined on a non-empty open set cannot be constant, 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 .
A map of the extended complex plane (which is conformally equivalent to a sphere) onto itself is conformal if and only if it is a Möbius transformation. Again, for the conjugate, angles are preserved, but orientation is reversed.
An example of the latter is taking the reciprocal of the conjugate, which corresponds to circle inversion with respect to the unit circle. This can also be expressed as taking the reciprocal of the radial coordinate in circular coordinates, keeping the angle the same. See also inversive geometry.
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.
Higher-dimensional Euclidean space
Any conformal map on a portion of Euclidean space of dimension greater than 2 can be composed from three types of transformation: a homothetic transformation, an isometry, and a special conformal transformation. (A special conformal transformation is the composition of a reflection and an inversion in a sphere.) Thus, the set of conformal transformations in spaces of dimension greater than 2 is much more restricted than in the planar case, where the Riemann mapping theorem provides a large set of conformal transformations.
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. See Conformal map projection.
Conformal mappings are invaluable for solving problems in engineering and physics that can be expressed in terms of functions of a complex variable but that 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 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, , 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) (see spherical wave transformation). 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
- 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. doi:10.1016/j.jsv.2013.09.002.
- 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", in Hazewinkel, Michiel (ed.), Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
- 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.
- Java applet by Jürgen Richter-Gebert using Cinderella.
- 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