# 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 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

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
, then this mapping is given by

where
is a constant, and
are the values, along the real axis of the
plane, of points corresponding to the vertices of the polygon in the
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 plane to one of the vertices of the plane polygon. By doing this, the first factor in the formula becomes constant and so can be absorbed into the constant . Conventionally, the point at infinity would be mapped to the vertex with angle .

## 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

Evaluation of this integral yields

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

This transformation is sketched below.

## Other simple mappings

### Triangle

A mapping to a plane triangle with interior angles and is given by

which can be expressed in terms of hypergeometric functions.

### Square

The upper half-plane is mapped to the square by

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)