%% Example Of A Locally Conformal Mapping: w = z^2, z ≠ 0
%% CSUMS Summer 2010 A.O. Hausknecht 
%% Written using OCTAVE
function plotZSqr_CircularGrid()
clf;

%% Create A Plot Of The Circular Grid
subplot (1, 2, 1)
title ("Grid Of Truncated Sectors")
hold on

%% Fill a truncated sector
%% 1. Generate points along the bottom edge 
%%    which is part of a circle.
t = linspace(pi/4,3*pi/8, 10);
x1 = 3*cos(t); y1 = 3*sin(t);
%%
%% 2. Generate points along the top edge 
%%    in revese order.
%%
%%    Reverese the order of the t-values.
t = fliplr (t);
%%    and generate the points
x2 = 4*cos(t); y2 = 4*sin(t);
%%
%% 3. Now collect all the points of the regions
%%    boundary together in counter-clockwise order!
x = cat(2, x1, x2); y = cat(2, y1,y2);
%%
%% 4. Finally, fill the region with gray
fill (x, y, [0.9 0.9 0.9])

%% Plot the grid's circles 
t = linspace(0,2*pi, 50);
for r =  1:4
  plot( r*cos(t),  r*sin(t),'b');
end

%% Plot the grid's radial lines
r = 0:.1:4;
for theta = 0: pi/8.0: 2*pi
  plot( r*cos(theta), r*sin(theta), 'r')
end
axis('equal'); axis([-5 5 -5 5])

%% Draw Unit Tangent vectors at z = (3+3i)/sqrt(2)
%% 1. Draw the tangent <-1, 1>/sqrt(2) to the circle at z
z = (3+3*i)/sqrt(2); dz = (-1+i)/sqrt(2);
a = [real(z) real(z + dz)]; b = [imag(z) imag(z + dz)];
line(a, b, 'linewidth',3, 'color', 'blue')
%%
%% 2. Draw the radial tangent <1, 1>/sqrt(2) to at z.
dz = (1+i)/sqrt(2);
a = [real(z) real(z + dz)]; b = [imag(z) imag(z + dz)];
line(a,b, 'linewidth',3, 'color', 'red')
hold off

%% Create A Plot The Image Of The Circular Grid
subplot (1, 2, 2)
title ("Image Of Grid Under z^2")
hold on

%% Fill the image of truncated sector
%% 1. Generate points along the bottom edge which is part of a circle.
t = linspace(pi/2,3*pi/4, 20);
x1 = 9*cos(t); y1 = 9*sin(t);
%%
%% 2. Generate points along the top edge  in revese order.
%%
%%    Reverese the order of the t-values.
t = fliplr (t);
x2 = 16*cos(t); y2 = 16*sin(t);
%%
%% 3. Now collect all the points of the regions
%%    boundary together in counter-clockwise order!
x = cat(2, x1, x2); y = cat(2, y1,y2);
%%
%% 4. Finally, fill the region with gray
fill (x, y, [0.9 0.9 0.9])


%% Plot images of the circles under w = z^2
t = linspace(0,2*pi, 50);
for r =  1:4
  plot(r*r*cos(t), r*r*sin(t),'b');
end

%% Plot images of the radial lines under w = z^2
rSqr = 0: .1: 16; 
for theta = 0: pi/4.0: 2*pi
  plot(rSqr*cos(theta), rSqr*sin(theta), 'r')
end
axis('equal')

%% Plot images of the Unit Tangent vectors at [ (3+3i)/sqrt(2)]^2
z = (3+3i)/sqrt(2); w = z^2;   
%% 1. Use the derivative to plot the image of the tangent perpindicular 
%%    to the circle: Note: dw = 2z*dz
dz = (-1+i)/sqrt(2); dw = 2*z*dz;
a1 = real(w);    b1 = imag(w);
a2 = real(w+dw); b2 = imag(w+dw); 
a = [a1 a2]; b = [b1 b2];
line(a,b, 'linewidth', 3, 'color', 'blue') 
%%
%% 2. Use the derivative to plot the image of the radial tangent  
%%    to the circle:  Note: dw = 2z*dz
dz = (1+i)/sqrt(2); dw = 2*z*dz;
a2 = real(w+dw); b2 = imag(w+dw);
a = [a1 a2]; b = [b1 b2];
line(a,b, 'linewidth', 3, 'color', 'red')

hold off