Schwarz–Christoffel mapping

In complex analysis, a Schwarz–Christoffel mapping is a conformal transformation of the upper half-plane onto the interior of a simple polygon. Schwarz–Christoffel mappings are used in potential theory and some of its applications, including minimal surfaces, hyperbolic art, and fluid dynamics. They are named after Elwin Bruno Christoffel and Hermann Amandus Schwarz.

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

${\displaystyle \{\zeta \in \mathbb {C}$ :\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 ${\displaystyle \alpha }$.

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

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

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

Other simple mappings

Triangle

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.

Square

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.

General triangle

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

See also

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

References

• Driscoll, Tobin A.; Trefethen, Lloyd N. (2002), Schwarz–Christoffel mapping, Cambridge Monographs on Applied and Computational Mathematics, 8, Cambridge University Press, doi:10.1017/CBO9780511546808, ISBN 978-0-521-80726-5, MR 1908657
• 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

Further reading

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).

External links

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