### Schwarz-Christoffel mapping
**My 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 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.
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. MATLAB is an
interactive high-level environment for numerical analysis and
scientific computing.The SC Toolbox exploits its interactivity and
powerful graphics, making the computation of conformal maps easier and
more flexible than ever before. In addition, the toolbox adds maps to
and from the half-plane, a strip, and a rectangle, and maps from a
disk to the exterior of a polygon.
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. Steve Vavasis and I wrote a paper describing a new algorithm for SC
mapping that avoids the problems normally encountered with
crowding. This algorithm is based on complex cross-ratios and Delaunay
triangulations, and it allows the solution of problems that are not
feasible for any other algorithm we know of. The new method is
incorporated into the latest version of the toolbox.
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. |