Schwarz–Christoffel mapping

In complex analysis, a Schwarz–Christoffel mapping is a conformal map of the upper half-plane or the complex unit disk onto the interior of a simple polygon.

Such a map is guaranteed to exist by the Riemann mapping theorem (stated by Bernhard Riemann in 1851); the Schwarz–Christoffel formula provides an explicit construction.

They were introduced independently by Elwin Christoffel in 1867 and Hermann Schwarz in 1869.

Schwarz–Christoffel mappings are used in potential theory and some of its applications, including minimal surfaces, hyperbolic art, and fluid dynamics.

The function f maps the real axis to the edges of the polygon.

Conventionally, the point at infinity would be mapped to the vertex with angle

values which generate the correct polygon side lengths.

This requires solving a set of nonlinear equations, and in most cases can only be done numerically.

This may be regarded as a limiting form of a triangle with vertices P = 0, Q = π i, and R (with R real), as R tends to infinity.

The upper half-plane is mapped to the square by where F is the incomplete elliptic integral of the first kind.

An analogue of SC mapping that works also for multiply-connected is presented in: Case, James (2008), "Breakthrough in Conformal Mapping" (PDF), SIAM News, 41 (1).