Conformal mapping of arc by bilinear function

%CONFORM1.M plots Figure 4.6, Conformal mapping of arc by bilinear function.
%       Conformal mapping of a circular arc by w = (z – a)/(z – b).
%       See Markushevitch, “Complex numbers and conformal mapping”.

alpha = pi/3: pi/90: 2*pi/3;
r = 2;
z = r*exp(i*alpha);
a = z(1);
b = z(length(z));
rad = [ 0 a ];
a1 = a + 1;
chord = [ b a1 ];
cx = real(chord);
cy = imag(chord);
t = [ (0 + i*r/cos(pi/6)) a ];
tx = real(t);
ty = imag(t);
w = (z – a*ones(size(z)))./(z – b*ones(size(z)));

subplot(1, 2, 1)
plot(real(z),imag(z),real(rad),imag(rad),real(a1),imag(a1),tx,ty,cx,cy)
axis(‘square’)
% axis([ -1 2 0 3 ])
title(‘z plane’)
xlabel(‘x’)
ylabel(‘y’)
text(1, 1.6, ‘a’)
text(-0.9, 1.6, ‘b’)
text(1.70, 1.6, ‘a1’)
text(real(t(1)), imag(t(1)), ‘t’)
subplot(1, 2, 2)
plot(real(w), imag(w), [ 0 10 ], [ 0 0 ])
axis(‘square’)
axis([ -25 14 -25 14 ])
title(‘w plane’)
xlabel(‘u’)
ylabel(‘v’)
text(-20,  8, ‘S’)
text(0, -4, ‘O’)
text(10, 0, ‘R’)
text(-20, -20, ‘w = (z – a)/(z – b)’)