Numerical conformal mapping software: zipper
ZIPPER finds a numerical approximation of the conformal map
(and its inverse) of the unit disk onto any Jordan region
(bounded or unbounded).
A potential drop between two boundary arcs(red),
with remaining arcs (green) insulated:
Incompressible, inviscid fluid flow (with circulation) around an object:
Conformal map of a Carleson grid on the disk to both the inside and the
outside of an inverted "snowflake".
Curve (blue) with least harmonic measure at 0 that meets every ray
from 0 to the unit circle.
The next six images give more detail on the
conformal map of an "inverted snowflake" onto the unit disk, and the
inverse of this map.
1. A rectangular grid inside the "inverted snowflake" (9K).
2. The image of the rectangular grid by a conformal map to the disk.
(3K)
3. Zooming in near the boundary of the disk for the previous picture.
(1K)
4. A standard Carleson grid on the disk. (8K)
5. The image of the Carleson grid by the conformal map to the
interior of the inverted snowflake and the image
by the conformal map to the exterior of the snowflake. (17K)
6. Zooming in near the boundary of the region for the previous picture.
(9K)
To use the conformal mapping programs, you will need a fortran
compiler. The X11-graphics for the demonstration program requires a C compiler.
If you do not have access to a unix or linnux machine, you will need
to find another graphics program to display the results of the
conformal mapping routines. It is possible to use gnuplot, which is
freely available. To obtain the source code (New version 6/07): click
here
(645K) then:
Save it as ``conformal.tar''
Then type:
- tar xf conformal.tar
- cd conformal/zipper
- more README_f (instructions for finding correct fortran makefile)
- make
- cd ../xcm
- more README_C (instructions for finding correct C makefile)
- make
- cd ..
- demo
- After you are familiar with demo, try the other
demonstration programs:
- demoresist (Electric Potential, resistance and extremal
length)
- demoflow (Fluid flow)
- demowhitney (Whitney decompositions and geodesics)
These instructions are repeated in the file conformal/README.
(Careful: your browser may create a file that appears to have text,
but you still must do the tar xf step above)
The demo allows you to create the boundary of a region graphically
by just clicking the mouse button. You can also input the boundary of
the region as a list of x,y coordinates of a finite number of boundary
points. If you don't have
access to X-11 graphics, send me a file with the x,y coordinates
of boundary points (two columns of numbers),
followed by one interior point in your region, and
I'll send you some pictures.
This material is based upon work
supported by the National Science Foundation, most recently under
Grant No. DMS-0602509.
Any opinions, findings, and conclusions or recommendations expressed
in this material are those of the author(s) and do not
necessarily reflect the views of the National Science Foundation.
Questions? Send them to me at:
marshall@math.washington.edu
Return to Don Marshall's home page.