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.
Consider a polygon in the complex plane. The Riemann mapping theorem implies that there is a biholomorphic mapping f from the upper half-plane
\{\zeta\inC:\operatorname{Im}\zeta>0\}
\alpha,\beta,\gamma,\ldots
f(\zeta)=\int\zeta
| K | |
| (w-a)1-(\alpha/\pi)(w-b)1-(\beta/\pi)(w-c)1-(\gamma/\pi) … |
dw
K
a<b<c< …
\zeta
z
The integral can be simplified by mapping the point at infinity of the
\zeta
z
K
\alpha
In practice, to find a mapping to a specific polygon one needs to find the
a<b<c< …
Consider a semi-infinite strip in the plane. This may be regarded as a limiting form of a triangle with vertices,, and (with real), as tends to infinity. Now and in the limit. Suppose we are looking for the mapping with,, and . Then is given by
f(\zeta)=\int\zeta
| K | |
| (w-1)1/2(w+1)1/2 |
dw.
Evaluation of this integral yields
z=f(\zeta)=C+K\operatorname{arcosh}\zeta,
where is a (complex) constant of integration. Requiring that and gives and . Hence the Schwarz–Christoffel mapping is given by
z=\operatorname{arcosh}\zeta.
This transformation is sketched below.
A mapping to a plane triangle with interior angles
\pia,\pib
\pi(1-a-b)
z=f(\zeta)=\int\zeta
| dw | |
| (w-1)1-a(w+1)1-b |
,
which can be expressed in terms of hypergeometric functions, more precisely incomplete beta functions.
The upper half-plane is mapped to the square by
z=f(\zeta)=\int\zeta
| dw | |
| \sqrt{w(1-w2) |
where F is the incomplete elliptic integral of the first kind.
The upper half-plane is mapped to a triangle with circular arcs for edges by the Schwarz triangle map.
An analogue of SC mapping that works also for multiply-connected is presented in: .