- Schwarz-Christoffel mapping
In
complex analysis , a discipline withinmathematics , a Schwarz-Christoffel mapping is a transformation of thecomplex plane that maps theupper half-plane conformally to apolygon . Schwarz-Christoffel mappings are used inpotential theory and some of its applications, includingminimal surface s andfluid dynamics . They are named afterElwin Bruno Christoffel andHermann Amandus Schwarz .In practice it is used in architecture and designing of aeroplanes.
Definition
Consider a polygon in the complex plane. The
Riemann mapping theorem implies that there is abijective holomorphic 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 interiorangle s , then this mapping is given by :where is aconstant , 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".It is often convenient to consider the case in which the
point at infinity of the plane maps to one of the vertices of the plane polygon (conventionally the vertex with angle ). If this is done, the first factor in the formula is effectively a constant and may be regarded as being absorbed into the constant .Example
Consider a semi-infinite strip in the math|z plane. This may be regarded as a limiting form of a triangle with vertices math|P = 0, math|Q = π i, and math|R (with math|R real), as math|R tends to infinity. Now math|α = 0 and math|β = γ = frac|π|2 in the limit. Suppose we are looking for the mapping math|f with math|f(−1) = Q, math|f(1) = P, and math|f(∞) = R. Then math|f is given by
:
Evaluation of this integral yields
:bigmath| z = f(ζ) = C + K macomp|ar cosh ζ
where math|C is a (complex) constant of integration. Requiring that math|f(−1) = Q and math|f(1) = P gives math|C = 0 and math|K = 1. Hence the Schwarz-Christoffel mapping is given by :bigmath|z = macomp|ar cosh ζThis transformation is sketched below.
Other simple mappings
Triangle
A mapping to a plane
triangle with angles and is given by:
quare
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 .ee also
* The
Schwarzian derivative appears in the theory of Schwarz-Christoffel mappings.References
* Tobin A. Driscoll and Lloyd N. Trefethen, "Schwarz-Christoffel Mapping," Cambridge University Press, 2002. ISBN 0-521-80726-3.
* Z. Nehari, "Conformal Mapping", (1952) McGraw-Hill, New York.Further reading
*.
External links
*
* [http://math.fullerton.edu/mathews/c2003/SchwarzChristoffelMod.html Schwarz-Christoffel Module by John H. Mathews]
* [http://www.timesonline.co.uk/tol/news/uk/science/article3478927.ece Eureka moment solves 140-year-old puzzle]
* [http://www.math.udel.edu/~driscoll/SC Schwarz-Christoffel toolbox] (software forMATLAB )
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