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====== Math 445 lecture 20: 3D graphics ======

Some examples of plotting 2d functions $f(x,y)$ in Matlab.

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==== 1d graphs ====


How would you make a graph of $f(x) = (x-1)^2$ on $-3 \leq x \leq 3$ in Matlab? You would
  - Create vector ''x'' of gridpoints with ''linspace''
  - Evaluate $f(x)$ on gridpoints using dot operations on ''x''
  - Use ''plot(x,f)'' to draw the graph

For example, 
<file matlab plot1d.m>
% Plot 1d graph of f(x) = (x-1)^2 on -3 < x < 3

x = linspace(-3, 3, 100); % create vector x of gridpoints
f = (x-1).^2;             % evaluate f(x) on gridpoints
plot(x,f)                 % plot  the graph

xlabel('x')
ylabel('f(x) = (x-1)^2')
</file>
produces
{{ :gibson:teaching:spring-2016:math445:onedplot.png?direct&400 |}}

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==== 2d graphs with meshgrid ====

Graphing a 2d function $f(x,y)$ is very similar. For example, to make a graph of the function 
$f(x,y) = 2*(x-1)^2 + (y-1)^2$ on $-4 \leq x \leq 4, -4 \leq y \leq 4$

  - Create vectors ''x'' and ''y'' of gridpoints on $x$ and $y$ axes with ''linspace''
  - Create matrices ''X'' and ''Y'' of gridpoints on 2d mesh with ''meshgrid''
  - Evaluate $f(x,y)$ on mesh of gridpoints using dot operations on ''X,Y''
  - Use ''pcolor(x,y,f)'' to draw the graph (or ''contour'', ''contourf'', ''surf'', ''surfc'')

For example,
<file matlab plot2d.m>
% Plot f(x,y) = 2*(x-1)^2 + (y-1)^2 on -4 < x < 4, -4 < y < 4

x = linspace(-4,4,30);     % create 1d gridpoints on x and y axes
y = linspace(-4,4,30);

[X,Y] = meshgrid(x,y);     % create 2d mesh of gridpoints over x,y

F = 2*(X-1).^2 + (Y-1).^2; % evaluate the function f(x,y) over mesh

pcolor(x,y,F);             % or contour(X,Y,F), surf(x,y,F), etc. 
colorbar()                 % plot a colorbar

xlabel('x');
ylabel('y');
title('f(x,y) = 2(x-1)^2 + (y-1)^2')

shading interp   % or faceted, flat
</file>

{{ :gibson:teaching:spring-2016:math445:lecture:twodplot.png?direct&400 |}}

----

==== quiver plots and the gradient===

The function $f(x,y) = x e^{-x^2 - y^2}$ has gradient

\begin{eqnarray*}
\nabla f = \left( \begin{array}{c} \partial f/\partial x \\  \partial f/\partial y \end{array} \right) = \left( \begin{array}{c} (1-x^2) e^{-x^2 - y^2} \\ -2xy e^{-x^2 - y^2} \end{array} \right)
\end{eqnarray*}

Here's the Matlab code to make a contour plot of $f(x,y)$ with its gradient superimposed as a quiver plot.

<file matlab plotgradient.m>
% Plot f(x,y) = x e^(-x^2 - y^2) with contours and its gradient as arrows.
x = linspace(-2, 2, 30);
y = linspace(-2, 2, 30);

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

F = X .* exp(-X.^2 - Y.^2);  % evaluate f(x,y) on mesh
contour(x,y,F);              % draw contour plot of f

% let dfdy = y component of grad F = df/dy

dfdx = (1-2*X.^2) .* exp(-X.^2 - Y.^2); % evaluate dfdx on mesh
dfdy = -2*X.*Y .* exp(-X.^2 - Y.^2);    % evaluate dfdy on mesh

hold on
quiver(x,y,dfdx,dfdy);       % draw quiver plot

xlabel('x')
ylabel('y')
title('f(x,y) = x e^{-x^2 - y^2} with \nabla f arrow plot')
colorbar()
</file>

{{ :gibson:teaching:spring-2016:math445:lecture:quiver.png?direct&500 |}}
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