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Schwarz-Christoffel mappingMy book, Schwarz-Christoffel Mapping with L. N. Trefethen, is now available from Cambridge University Press.
A conformal map of a region in the complex plane is an
analytic (smooth) function whose derivative never vanishes within the
region. Graphically, a
In the late 1970's Nick Trefethen publicly released the Fortran package SCPACK, which is a fast and reliable tool for maps between the unit disk and a polygon. Subsequently, I have written (with Trefethen's blessing) the natural successor to SCPACK, the Schwarz-Christoffel Toolbox for MATLAB.
You can read a detailed overview of an older version of the SC Toolbox, which appeared in the June 1996 ACM Transactions on Mathematical Software. You can also download the toolbox. It runs best under version 6 of MATLAB (though an older version for MATLAB 4.1 still exists). In conjunction with Doug Arnold, who has put together a wonderful page of graphics for complex analysis, I have created a few animations illustrating Schwarz-Christoffel maps. |
conformal map transforms any pair of
curves intersecting at a point in the region so that the image curves
intersect at the same angle. Conformal maps are both mathematically
interesting and practically useful. Because of the preservation of
angles, a conformal map of a square grid in the plane results in a
curvilinear orthogonal grid. Furthermore, the transplantation by a
conformal map of a solution to Laplace's equation is still a solution
in the image region, so that conformal maps can be used in heat
transfer, electrostatics, steady fluid flows, and other phyiscal
applications in which Laplace's equation arises.
In general
it's impossible to write down a simple formula for the conformal map
from one region to another. But for the important special case of a
map from the upper half-plane (Im z > 0) to a polygon, there is
such a formula, discovered independently by Schwarz and Christoffel in
the 1860's. There are two catches, however. The first is that except
for some very simple polygons, the formula requires an integral that
has no closed form. The second, more fundamental catch is that you
first need to find out the values of some unspecified constants in the
formula. These constants are the "pre-images" of the vertices of the
polygon on the real axis. To find these prevertices, you need
to solve a set of highly nonlinear equations. Both of these issues can
be surmounted by using a computer.
A
limitation of every traditional numerical conformal mapping method
goes by the name of crowding. This occurs whenever the target
region has elongated or pinched regions.