Schwarz–Christoffel mapping

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Conformal mapping in complex analysis

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^[1] and Hermann Schwarz in 1869.^[2]

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

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

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.^[3]

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.

A mapping to a plane triangle with interior angles ${\displaystyle \pi a,\,\pi b}$ and ${\displaystyle \pi (1-a-b)}$ is given by

${\displaystyle z=f(\zeta )=\int ^{\zeta }{\frac {dw}{(w-1)^{1-a}(w+1)^{1-b}}},}$

which can be expressed in terms of hypergeometric functions or incomplete beta functions.

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

The upper half-plane is mapped to the square by

${\displaystyle z=f(\zeta )=\int ^{\zeta }{\frac {\mathrm {d} w}{\sqrt {w(1-w^{2})}}}={\sqrt {2}}\,F\left({\sqrt {\zeta +1}};{\sqrt {2}}/2\right),}$

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

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

• Driscoll, Tobin A.; Trefethen, Lloyd N. (2002). Schwarz–Christoffel Mapping. Cambridge University Press. doi:10.1017/CBO9780511546808. ISBN 9780521807265.
• 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)