%% Example Of A Locally Conformal Mapping: w = 1/z, z ≠ 0
%% CSUMS Summer 2010 A.O. Hausknecht 
%% Written using OCTAVE
function plotZInverse_CircularGrid()
clf;

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

%% Draw filled of truncated sectors
plotFilledSector(0*pi/4, 1, 1*pi/4, 2, [0.9 0.9 0.9])
plotFilledSector(1*pi/4, 1, 2*pi/4, 2, [1 0 0])
plotFilledSector(2*pi/4, 1, 3*pi/4, 2, [0 1 0])
plotFilledSector(3*pi/4, 1, 4*pi/4, 2, [0 0 1])

plotFilledSector(0*pi/4, 0.5, 1*pi/4, 1, [1 1 0])
plotFilledSector(1*pi/4, 0.5, 2*pi/4, 1, [0 1 1])
plotFilledSector(2*pi/4, 0.5, 3*pi/4, 1, [1 0 1])
plotFilledSector(3*pi/4, 0.5, 4*pi/4, 1, [0.5 0.5 1])

%% Plot the grid's circles 
t = linspace(0,2*pi, 50);
for r =  [0.5 2]
  plot( r*cos(t),  r*sin(t),'b');
end
plot( cos(t),  sin(t),'k', 'linewidth', 3);

%% Plot the grid's radial lines
r = 0:.1:2;
for theta = 0: pi/4.0: 2*pi
  plot( r*cos(theta), r*sin(theta), 'r')
end
%% Set the axes domain, range, and scaling.
axis('equal')
axis([-3 3 -3 3])

%% Plot Unit Tangents
plotUnitTangents(2 ,  'blue', 'yellow')
plotUnitTangents((2+2i)/sqrt(2) , 'green', 'cyan')

hold off

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

%% Draw images of filled truncated sectors
plotImageOfFilledSector(0*pi/4, 1, 1*pi/4, 2, [0.9 0.9 0.9])
plotImageOfFilledSector(1*pi/4, 1, 2*pi/4, 2, [1 0 0])
plotImageOfFilledSector(2*pi/4, 1, 3*pi/4, 2, [0 1 0])
plotImageOfFilledSector(3*pi/4, 1, 4*pi/4, 2, [0 0 1])

plotImageOfFilledSector(0*pi/4, 0.5, 1*pi/4, 1, [1 1 0])
plotImageOfFilledSector(1*pi/4, 0.5, 2*pi/4, 1, [0 1 1])
plotImageOfFilledSector(2*pi/4, 0.5, 3*pi/4, 1, [1 0 1])
plotImageOfFilledSector(3*pi/4, 0.5, 4*pi/4, 1, [0.5 0.5 1])


%% Plot images of the circles under w = 1/z
t = linspace(0,2*pi, 50);
for r =  [0.5 2]
  plot(cos(t)/r, -sin(t)/r,'b');
end
plot( cos(t),  -sin(t),'k', 'linewidth', 3);

%% Plot images of the radial lines under w = z^2
r = 0.5: .1: 2; 
for theta = 0: pi/4.0: 2*pi
  plot(r*cos(theta), r*sin(theta), 'r')
end
%% Set the axes domain, range, and scaling.
axis('equal')
axis([-3 3 -3 3])

%% Plot the images of the Unit Tangents
plotImagesOfUnitTangents(2 ,  'blue', 'yellow')
plotImagesOfUnitTangents((2+2i)/sqrt(2) , 'green', 'cyaN')

hold off
end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Helper functions used above

function plotFilledSector(t0, r0, t1, r1, color)
%% Fill a truncated sector
%% 1. Generate points along the bottom edge 
%%    which is part of a circle.
t = linspace(t0, t1, 10);
x1 = r0*cos(t); y1 = r0*sin(t);
%%
%% 2. Generate points along the top edge 
%%    in revese order and generate the points
t = fliplr (t);
x2 = r1*cos(t); y2 = r1*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
fill (x, y, color)
end

function plotImageOfFilledSector(t0, r0, t1, r1, color)
%% Fill the image of truncated sector
%% 1. Generate points along the bottom edge 
%%    which is part of a circle.
t = linspace(t0, t1, 20);
x1 = cos(t)/r0; y1 = -sin(t)/r0;
%%
%% 2. Generate points along the top edge 
%%    in revese order.
t = fliplr (t);
x2 = cos(t)/r1; y2 = -sin(t)/r1;
%%
%% 3. Now collect all the points of the region's
%%    boundary together in counter-clockwise order!
x = cat(2, x2, x1); y = cat(2, y2, y1);
%%
%% 4. Finally, fill the region with green
fill (x, y, color)
end

function plotUnitTangents(z, color1, color2)
%% Draw Unit Tangent vectors at z
%%
%% 1. Draw the radial tangent to at z.
dz = z/abs(z);
a = [real(z) real(z + dz)]; b = [imag(z) imag(z + dz)];
line(a, b, 'linewidth',3, 'color', color1);
%%
%% 2. Draw the tangent dz to the circle at z.
%%    Note: Use J(x+yi) = -y+xi to obtain perpindicular vector
dz = -imag(dz) + real(dz)*i;
a = [real(z) real(z + dz)]; b = [imag(z) imag(z + dz)];
line(a,b, 'linewidth',3, 'color', color2)
end

function plotImagesOfUnitTangents(z, color1, color2)
%% Plot images of the Unit Tangent vectors at w = 1/z
w = 1/z; 
%% 1. Use the derivative to plot the image of the radial tangent  
%%    to the circle:  Note: dw = -1/z^2*dz
dz = z/abs(z); dw = -1/z^2*dz;
a = [real(w)  real(w+dw)]; b = [imag(w) imag(w+dw)];
line(a,b, 'linewidth', 3, 'color', color1);
%%
%% 2. Draw the tangent dz to the circle at z.
%%    Note: Use J(x+yi) = -y+xi to obtain perpindicular vector
dz = -imag(dz) + real(dz)*i; dw = -1/z^2*dz;
a = [real(w)  real(w+dw)]; b = [imag(w) imag(w+dw)];
line(a,b, 'linewidth', 3, 'color', color2);
end