%% Example of a Globally Conformal Mapping: w = Az, A ≠ 0
%% CSUMS Summer 2010 A.O. Hausknecht 
%% Written using OCTAVE
function plotAz_SquareGrid(A=0.5)
printf("A = %g\n", A)
clf;

%% Default grid parameters
minXY = -4; maxXY = 4; n = 16;
delta = (maxXY-minXY)/n;

%% Create A Plot Of The Rectangular Grid
subplot (1, 2, 1)
title ("Grid of Squares")
hold on

%% Draw left-top filled square
x1 = maxXY-1; x2 = maxXY;
rectX = [x1 x2 x2 x1]; 
y1 = x1; y2 = x2;
rectY = [y1 y1 y2 y2];
%% Fill with gray
fill (rectX, rectY, [0.9 0.9 0.9])

%% Draw the vertical lines
y = minXY:delta:maxXY;
for x =  minXY:delta:maxXY
  plot( x*ones(1, n+1) , y,'b')
end

%% Draw the horizontal lines
x = minXY:delta:maxXY;
for y = minXY:delta:maxXY
  plot(x, y*ones(1, n+1), 'r')
end
plot(0, 0, 'o', 'markersize', 4, 'color', 'k')
%%
%% Set the axes scaling, domain, and range
axis('equal')
rMax = max([maxXY*sqrt(2)*abs(A), maxXY*sqrt(2)]);
axis([-rMax rMax -rMax rMax])

%% Draw Unit Tangents at z = x1+y1*i
%% 1. Draw the Unit Tangent in the i-direction
tx = [x1 x1]; ty = [y1 (y1+1)];
line(tx, ty, 'linewidth',3, 'color', 'blue')
%% 1. Draw the Unit Tangent in the r-direction
tx = [x1 (x1+1)]; ty = [y1 y1];
line(tx, ty, 'linewidth',3, 'color', 'red')
hold off 

%% Create A Plot Of The Image Of The Rectangular Grid
subplot (1, 2, 2)

str = strcat("Image of Grid Under w = (", num2str (A, "%5.2f"), ")z");
title (str)
hold on

%% Draw the image of a square under w = Az
%% Need to form a list of x-values and y-values
%% along the image's boundary in counter-clockwise order.
y = minXY:delta: maxXY; xy = maxXY-1; t = xy:.1:maxXY;
%% Bottom edge's image
w = A*(t + xy*i);    u1 = real(w); v1 = imag(w);
%% Right edge's image
w = A*(maxXY + t*i); u2 = real(w); v2 = imag(w);
%%
%% Revere the order of the t-values so that the points on
%% the top and left edges are generated in counter-clockwise order.
t = fliplr (t);
%% Top edge's image
w = A*(t + maxXY*i); u3 = real(w); v3 = imag(w);
%% Left edge's image
w = A*(xy + t*i); u4 = real(w); v4 = imag(w);
%%
%% Now collect all the points in the correct order!
u = cat(2, u1, u2, u3, u4);
v = cat(2, v1, v2, v3, v4);
%% Finally, fill the image of the square
fill (u, v, [0.9 0.9 0.9])

%% Draw the images of the vertical lines
y = minXY: maxXY;
for x =  minXY:delta:maxXY
  w = A*(x+y*i); plot(real(w), imag(w),'b')
end

%% Draw the images of the horizontal lines
x = minXY:delta: maxXY;
for y = minXY:delta:maxXY
  w = A*(x+y*i); plot(real(w), imag(w),'r')
end
plot(0, 0, 'o', 'markersize', 4, 'color', 'k')
%% Set the axes scaling, domain, and range
axis('equal')
axis([-rMax rMax -rMax rMax])

%% Draw the images of the unit tangents under w = Az
%% Note: dw = Adz,
%% 
%% 1. Plot The image of the unit tangent in the i-direction
z = xy+xy*i; dz = 0+1*i;  
w = A*z; dw = A*dz;
tx = [real(w) real(w+dw)]; ty = [imag(w) imag(w+dw)];
line(tx, ty, 'linewidth',3, 'color', 'blue')
%%
%% 2. Plot The image of the unit tangent in the real-direction
dz = 1 + 0*i; dw = A*dz; 
tx = [real(w) real(w+dw)]; ty = [imag(w) imag(w+dw)];
line(tx, ty, 'linewidth',3, 'color', 'red')

hold off 
end