Schwarz–Christoffel mapping

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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 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 bijective biholomorphic mapping f from the upper half-plane

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

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

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

where $K$ is a constant, and $a<b<c<...$ are the values, along the real axis of the $\zeta$ 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 $\zeta$ plane maps to one of the vertices of the $z$ plane polygon (conventionally the vertex with angle $\alpha$ ). 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 $<var>z</var>$ plane. This may be regarded as a limiting form of a triangle with vertices $<var>P</var> = 0$ , $<var>Q</var> = π i$ , and $<var>R</var>$ (with $<var>R</var>$ real), as $<var>R</var>$ tends to infinity. Now $α = 0$ and $β = γ = π ⁄ 2$ in the limit. Suppose we are looking for the mapping $<var>f</var>$ with $<var>f</var>(- 1) = <var>Q</var>$ , $<var>f</var>(1) = <var>P</var>$ , and $<var>f</var>(\infty) = <var>R</var>$ . Then $<var>f</var>$ is given by

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

Evaluation of this integral yields

z = f (ζ) = C + K arccosh ζ

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

z = arccosh ζ

This transformation is sketched below.

Other simple mappings

Triangle

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

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

Square

The upper half-plane is mapped to the square by

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

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.

References

Driscoll, Tobin A.; Trefethen, Lloyd N. (2002), Schwarz-Christoffel mapping, Cambridge Monographs on Applied and Computational Mathematics 8, Cambridge University Press, 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

External links

Schwarz-Christoffel transformation, PlanetMath.org.
Schwarz–Christoffel Module by John H. Mathews
Schwarz–Christoffel toolbox (software for MATLAB)

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