%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Octave (or MATLAB) commands to graph
% intersecting lines and planes.
%
% Author: M.V.Wickerhauser
% Date: 2020-08-09
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%%%%%%%%%%%%%%%%
% PLOTTING LINES
%
% Suppose we have two general line equations:
%
%  -2*x + 3*y =  1
%    -x - 2*y = -2
% 
% To graph these in Octave (or MATLAB) we must first
% decide how to write one of the variables in terms of
% the other two, since the software requires the form
% y=f(x) with x specified and y computed, or else x=f(y).
% For this example, we choose to solve for y to get the
% slope-intercept form y=f(x)=m*x+b.  This gives us
%  
%  y =  (2/3)*x + 1/3 = f1(x)
%  y =  (-1/2)*x + 1  = f2(x)
%
% Next, we choose a grid of x points at which to evaluate
% the y values f1, f2:
x = linspace (-5, 5, 101)';  % 101 equispaced points in [-5,5]

% Evaluate f1, f2 to get the two lines' y-coordinates:
f1 =  (2/3)*x + 1/3;
f2 =  (-1/2)*x + 1;

% Plot multiple graphs on the same figure using "hold on" 
clf;       % [cl]ear any previous [f]igure
plot(x,f1,color="k");   % first line in blac[k]
hold on;                % superpose subsequent graphs rather than erase
plot(x,f2,color="r");   % second line in [r]ed

% NOTE: the intersection point is [4/7 5/7] 


%%%%%%%%%%%%%%%%%
% PLOTTING PLANES
%
% Suppose we have three general plane equations:
%
%  -2*x + 3*y + z =  1
%    -x - 2*y + z = -2
%   3*x -   y + z =  4
%
% To graph these in Octave (or MATLAB) we must first
% decide how to write one of the variables in terms of
% the other two, since the software requires the form
% z=f(x,y) with x,y specified and z computed, or else
% y=f(x,z), or x=f(y,z).  For this example, it is natural
% to solve for z since z appears with coefficient 1 in all
% three equations.  This gives us
%  
%  z =  2*x - 3*y + 1 = f1(x,y)
%  z =    x + 2*y - 2 = f2(x,y)
%  z = -3*x +   y + 4 = f3(x,y)
%
% Next, we choose a grid of x,y points at which to evaluate
% the z values f1, f2, f3:
tx = linspace (-8, 8, 41)';  % 41 equispaced points in [-8,8]
ty = tx;  % Same for y.  Note that these are column vectors.

% Now create two 41x41 matrices of the x and y coordinates
[x, y] = meshgrid (tx, ty);

% Evaluate f1, f2, f3 to get the three planes' z-coordinates:
f1 =  2*x - 3*y + 1;
f2 =    x + 2*y - 2;
f3 = -3*x +   y + 4;

% Plot in three dimensions using "mesh()"
% Plot multiple graphs on the same figure using "hold on" 
clf;       % [cl]ear any previous [f]igure
mesh(tx,ty,f1);   % first plane
hold on;          % superpose subsequent graphs rather than erase
mesh(tx,ty,f2);   % second plane
mesh(tx,ty,f3);   % third plane

% Change the colors.  
colormap hot     % red, orange, yellow
colormap cool    % blue, green, magenta
colormap list    % print a list of predefined colormap names.

% NOTE: the intersection is a single point [9/7, 6/7, 1]