# Schwarz-Christoffel toolbox and Chebfun

Nick Trefethen, October 2010

## Contents

(Chebfun example complex/SCToolbox.m)

Chebfun's SCRIBBLE command is good for illustrating conformal maps in the complex plane. These maps can distort distances greatly but preserve all angles, as we can demonstrate by mapping English text.

## Tanh map

The hyperbolic tangent function maps infinite strips onto lens-shaped regions with vertices at -1 and 1. We use SCRIBBLE to define a piecewise smooth parameterized curve whose smooth segments describe letter shapes in the complex plane. The letters are scaled and translated using complex arithmetic.

w = .07-.15i + 2.2*scribble('INFINITE STRIP');


We also use chebfuns to represent the two sides of the strip, and plot everything together.

bndry = chebfun('1i*pi/8 + t',[-3 3]);
bndry = [bndry; -bndry];
figure('defaultlinelinewidth',1.8)
plot(w), hold on
plot(bndry,'k'), axis equal, axis([-3 3 -1.5 1.5]) Here, we simply repeat the plots after composing each element with the tanh function.

g = @(z) tanh(z);
clf, plot(g(w)), hold on
plot(g(bndry),'k'), axis equal, axis([-1.4 1.4 -1 1]) The tanh map is the basis of numerical methods based on sinc functions, discussed in books and papers by F. Stenger [4,5]; see also .

## Schwarz-Christoffel maps

More generally, suppose we want to map the original infinite strip not to the lens shape but to a polygon. This kind of map is provided by the Schwarz-Christoffel formula , which is implemented numerically in Driscoll's Schwarz-Christoffel Toolbox . Here is an illustration of the map to a rectangle.

As before, we set up the letters and the boundary of the strip.

w = .07+0.3i + 3*scribble('INFINITE STRIP');
bndry = [chebfun('1i + t',5*[-1 1]); chebfun('t',5*[-1 1])];


(The SC Toolbox must be on MATLAB's path for the followingto work.)

if ~exist('scgui','file'), return, end


Next, define the map to a specified rectangle such that the ends of the strip map to two corners. A few extra manipulations are done to make a map that works for our purposes.

p = polygon([-1 -0.5 0 0.5 1 1+.5i -1+.5i]);     % rectangle
f = stripmap(p,[1 5]);                           % map strip -> rectangle
z = prevertex(f); g = stripmap(p,z-z(3),[1 5]);  % renormalized map


Here is where we compose the SC map with the letters.

gw = chebfun(@(x) g(w(x)),w.ends,'eps',1e-4,'extrapolate','on');


Finally, we plot everything.

clf, subplot(2,1,1)
plot(w), axis equal
hold on, plot(bndry,'k'), xlim(4.5*[-1,1])
subplot(2,1,2)
plot(gw), axis equal
vp = vertex(p); vp = vp([1:end 1]);
hold on, plot(vp,'k'), xlim(1.5*[-1 1]) References:

 T. A. Driscoll, Algorithm 843: Improvements to the Schwarz-Christoffel Toolbox for MATLAB, ACM Transactions on Mathematical Software 31 (2005), 239-251.

 T. A. Driscoll and L. N. Trefethen, Schwarz-Christoffel Mapping, Cambridge U. Press, 2002.

 M. Richardson and L. N. Trefethen, A sinc function analogue of Chebfun, SIAM Journal on Scientific Computing 33 (2011), 2519-2535.

 F. Stenger, Numerical Methods Based on Sinc and Analytic Functions, Springer, 1993.

 F. Stenger, Handbook of Sinc Numerical Methods, CRC Press, 2010.