% airfoil.m

clear; clc
syms x y real; z=x+1i*y;
syms t real

Joukowsky = @(z) 1/2*(z+1/z);
w=Joukowsky(z),  X=real(w),  Y=imag(w)

%% unitary circle
circ = exp(1i*t);

%% circle passing throug 1 amd centered at z0
z0=-0.1+0.2i; % center
r=abs(1-z0);  % radius (passa per 1)
circ0=z0+r*circ;

figure(1); clf
    h=ezplot(real(circ),imag(circ),[-pi pi]); set(h,'Color','k','Linewidth',2)
    hold on; grid
    h=plot(real(z0),imag(z0),'or'); set(h,'MarkerFaceColor','r')
    plot([real(z0) 1],[imag(z0) 0],':b')
    h=ezplot(real(circ0),imag(circ0),[-pi pi]); set(h,'Color','b','Linewidth',2)
    axis equal; axis([-1.5 1.5 -1.5 1.5])
    title(['\Gamma = \Gamma(' num2str(z0) ',' num2str(r) ')'],'Color','b','FontSize',18,'FontWeight','normal');
    xlabel('real(z)'); ylabel('imag(z)')

%% images by means of Joukowsky map
wcirc = Joukowsky(circ);
wcirc0= Joukowsky(circ0);

figure(2); clf
    h=ezplot(real(wcirc),imag(wcirc),[-pi pi]); set(h,'Color','k','Linewidth',2)
    hold on; grid on
    h=ezplot(real(wcirc0),imag(wcirc0),[-pi pi]); set(h,'Color','b','Linewidth',2)
    axis equal; axis([-1.2 1.2 -1.2 1.2])
    title('image of \Gamma by Joukowsky map: w = (z+1/z)/2','Color','b','FontSize',18,'FontWeight','normal');
    xlabel('real(w)'); ylabel('imag(w)')