Schwarz–Christoffel mapping

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

${\displaystyle \{\zeta \in \mathbb {C}$ :\operatorname {Im} \zeta >0\}}

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 ${\displaystyle \alpha ,\beta ,\gamma ,\ldots }$, then this mapping is given by

${\displaystyle f(\zeta )=\int ^{\zeta }{\frac {K}{(w-a)^{1-(\alpha /\pi )}(w-b)^{1-(\beta /\pi )}(w-c)^{1-(\gamma /\pi )}\cdots }}\,\mathrm {d} w}$

where ${\displaystyle K}$ is a constant, and ${\displaystyle a<b<c<\cdots }$ are the values, along the real axis of the ${\displaystyle \zeta }$ plane, of points corresponding to the vertices of the polygon in the ${\displaystyle z}$ plane. A transformation of this form is called a Schwarz–Christoffel mapping.

The integral can be simplified by mapping the point at infinity of the ${\displaystyle \zeta }$ plane to one of the vertices of the ${\displaystyle z}$ plane polygon. By doing this, the first factor in the formula becomes constant and so can be absorbed into the constant ${\displaystyle K}$. Conventionally, the point at infinity would be mapped to the vertex with angle ${\displaystyle \alpha }$.

In practice, to find a mapping to a specific polygon one needs to find the ${\displaystyle a<b<c<\cdots }$ 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.[1]

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

${\displaystyle f(\zeta )=\int ^{\zeta }{\frac {K}{(w-1)^{1/2}(w+1)^{1/2}}}\,\mathrm {d} w.\,}$

Evaluation of this integral yields

${\displaystyle z=f(\zeta )=C+K\operatorname {arcosh} \zeta ,}$

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

${\displaystyle z=\operatorname {arcosh} \zeta .}$

This transformation is sketched below.

Schwarz–Christoffel mapping of the upper half-plane to the semi-infinite strip

• The Schwarzian derivative appears in the theory of Schwarz–Christoffel mappings.

• Christoffel, Elwin Bruno (1867). "Sul problema delle temperature stazionarie e la rappresentazione di una data superficie" [On the problem of stationary temperatures and the representation of a given surface]. Annali di Matematica Pura ed Applicata (in Italian). 1: 89–103. doi:10.1007/BF02419161. S2CID 121089696.
• Driscoll, Tobin A.; Trefethen, Lloyd N. (2002). Schwarz–Christoffel Mapping. Cambridge University Press. doi:10.1017/CBO9780511546808. ISBN 9780521807265.
• Schwarz, Hermann Amandus (1869). "Ueber einige Abbildungsaufgaben" [About some mapping problems]. Crelle's Journal (in German). 1869 (70): 105–120. doi:10.1515/crll.1869.70.105. S2CID 121291546.
• Forsyth, Andrew Russell (1918) [1st ed. 1893]. Theory of Functions of a Complex Variable. Cambridge. §§267–270, pp. 665–677.
• Nehari, Zeev (1982) [1952], Conformal mapping, New York: Dover Publications, ISBN 978-0-486-61137-2, MR 0045823
• The Conformal Hyperbolic Square and Its Ilk Chamberlain Fong, Bridges Finland Conference Proceedings, 2016

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).

• "Schwarz–Christoffel transformation". PlanetMath.
• Schwarz–Christoffel toolbox (software for MATLAB)