====== Math 445: 3D graphics ======

Matlab vocabulary
  * meshgrid
  * pcolor
  * contour, contourf
  * surf, surfc
  * quiver
  * shading (flat, faceted, or interp)
  * colorbar
  * axis (equal, tight)
  
===== meshgrid: =====

The ''meshgrid'' function is essential for Matlab's 3D graphics. Meshgrid creates 2D arrays of x,y data covering the x,y, plane, over which a function can be evaluated and graphed. Example:

<code matlab>
>> x = linspace(-2,2,5) 

x =
    -2    -1     0     1     2

>> y = linspace(-3,3,7)

y =
    -3    -2    -1     0     1     2     3

>> [X,Y] = meshgrid(x,y)

X =
    -2    -1     0     1     2
    -2    -1     0     1     2
    -2    -1     0     1     2
    -2    -1     0     1     2
    -2    -1     0     1     2
    -2    -1     0     1     2
    -2    -1     0     1     2

Y =
    -3    -3    -3    -3    -3
    -2    -2    -2    -2    -2
    -1    -1    -1    -1    -1
     0     0     0     0     0
     1     1     1     1     1
     2     2     2     2     2
     3     3     3     3     3
</code>
It is conventional to use capital letters (X,Y) for the matrix output of meshgrid from small letter (x,y) vector inputs. Observe how the output matrices X varies left to right, along the x axis, and Y varies up and down, along the y axis. Together they provide x,y, coordinates for a grid of points on the x,y plane in the region $-2 \leq x \leq 2$, $-3 \leq y \leq 3$.
 
We can then evaluate a function $z = f(x,y)$ over that 2D array via elementwise matrix operations. For example, this Matlab code would evaluate $z = f(x,y) = x^2 + y^2$

<code matlab>
Z = X.^2 + Y.^2
</code>

===== pcolor: pseudocolor plot =====

The ''pcolor'' function produces a pseudocolor or checkerboard plot of Z as a function of x,y.
<code matlab>
pcolor(X,Y,Z) 
colorbar
axis equal 
axis tight
xlabel('x'); ylabel('y'); title('z = x^2 + y^2')
</code>
You can modify the appearance of the pseudocolor with the ''shading'' command. Try
''shading flat'', ''shading interp'', and ''shading faceted''.

===== contour, contourf =====

The ''contour'' function plots contours or level curves of z=f(x,y). That is, it plots curves
on which f(x,y) is constant.

<code matlab>
[X,Y] = meshgrid(x,y);
Z = X.^2 + Y.^2;
contour(X,Y,Z)
xlabel('x'); ylabel('y'); title('z= x^2 + y^2')
colorbar
axis equal 
axis tight
</code>
As you can see, the level curves of $z = f(x,y) = x^2 + y^2$ are circles $x^2 + y^2 = c$. 

''contourf'' is the same as ''contour'', except that the regions between contour lines are filled with color. 

===== surf, surfc =====

The previous plots were all looking straight down at the (x,y) plane, with the value of z = f(x,y) encoded as a color. The ''surf'' function will plot z = f(x,y) in 3D, as a surface of height z over the (x,y) plane. 

<code matlab>
surf(X,Y,Z) % draw z=f(x,y) as a surface over x,y
xlabel('x'); ylabel('y'); zlabel('z')
axis equal; axis tight
</code>

It's also possible to draw more complicated surfaces (surfaces that are not simple graphs of the form
$z = f(x,y)$). Here's an example of how to draw a sphere by parameterizing its surface in terms of angles $\phi$ and $\theta$. 

<code matlab>
% make mesh over theta, phi
theta = linspace(0,2*pi,50); % angle between x and y
phi   = linspace(0,pi,25);   % angle down from z axis
 
[Theta, Phi] = meshgrid(theta, phi); % form 2D mesh in theta, phi

% Parameterize surface of sphere in terms of theta, phi
% (note that x^2 + y^2 + z^2 = 1)
X = cos(Theta).*sin(Phi);
Y = sin(Theta).*sin(Phi);
Z = cos(Phi);     

% Draw parametrized surface of sphere with surf
surf(X,Y,Z); axis equal
xlabel('x'); ylabel('y'); zlabel('z')
axis equal
</code>