Schwarz–Christoffel mapping

Definition

Consider a polygon in the complex plane. The Riemann mapping theorem implies that there is a biholomorphic mapping f from the upper half-plane

to the interior of the polygon. The function f maps the real axis to the edges of the polygon. If the polygon has interior angles α , β , γ , … , then this mapping is given by

where K is a constant, and a < b < c < ⋯ are the values, along the real axis of the ζ plane, of points corresponding to the vertices of the polygon in the z plane. A transformation of this form is called a Schwarz–Christoffel mapping.

It is often convenient to consider the case in which the point at infinity of the ζ plane maps to one of the vertices of the z plane polygon (conventionally the vertex with angle α ). If this is done, the first factor in the formula is effectively a constant and may be regarded as being absorbed into the constant K .

Example

Consider a semi-infinite strip in the z plane. 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. Now α = 0 and β = γ =  ^π⁄₂ in the limit. Suppose we are looking for the mapping f with f(−1) = Q, f(1) = P, and f(∞) = R. Then f is given by

Evaluation of this integral yields

where C is a (complex) constant of integration. Requiring that f(−1) = Q and f(1) = P gives C = 0 and K = 1. Hence the Schwarz–Christoffel mapping is given by

This transformation is sketched below.

Triangle

A mapping to a plane triangle with angles π a , π b and π ( 1 − a − b ) is given by

Square

The upper half-plane is mapped to the square by

where F is the incomplete elliptic integral of the first kind.

General triangle

The upper half-plane is mapped to a triangle with circular arcs for edges by the Schwarz triangle map.