Conformal mapping

(mathematics) An angle-preserving analytic function of a complex variable.

A special operation in mathematics in which a set of points in one coordinate system is mapped or transformed into a corresponding set in another coordinate system, preserving the angle of intersection between pairs of curves.

A mapping or transformation of a set E of points in the xy plane onto a set F in the uv plane is a correspondence that is defined for each point (x, y) in E and sends it to a point (u, v) in F, so that each point in F is the image of some point in E. A mapping is one to one if distinct points in E are transformed to distinct points in F. A mapping is conformal if it is one to one and it preserves the magnitudes and orientations of the angles between curves. Conformal mappings preserve the shape but not the size of small figures.

If the points (x, y) and (u, v) are viewed as the complex numbers z = x + iy and w = u + iv, the mapping becomes a function of a complex variable: w = f(z). It is an important fact that a one-to-one mapping is conformal if and only if the function f is analytic and its derivative f′(z) is never equal to zero.

Conformal mappings are important in two-dimensional problems of fluid flow, heat conduction, and potential theory. They provide suitable changes of coordinates for the analysis of difficult problems. For example, the problem of finding the steady-state distribution of temperature in a conducting plate requires the calculation of a harmonic function with prescribed boundary values. If the region can be mapped conformally onto the unit disk, the transformed problem is readily solved by the Poisson integral formula, and the required solution is the composition of the resulting harmonic function with the conformal mapping. The method works because a harmonic function of an analytic function is always harmonic. See also Conduction (heat); Potentials.

The term conformal applies in a more general context to the mapping of any surface onto another. A problem of great importance for navigation is to produce conformal mappings of a portion of the Earth's surface onto a portion of the plane. The Mercator and stereographic projections are conformal in this sense. See also Map projections.