Plotting Functions of Two Variables

Section 3.5 Plotting Functions of Two Variables

Subsection 3.5.1 Functions on Rectangular Grids

Suppose you wish to plot a function $f(x,y)$ on the rectangle $a \le x \le b$ and $c \le y \le d\text{.}$ The graph of a function of two variables is of course a three dimensional object. Visualizing the graph is often very useful.

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For example, suppose you have a formula

\begin{equation*} f(x,y) = x \sin(xy) \end{equation*}

and you are interested in the function on the region $0 \le x \le 5\text{,}$ $\pi \le y \le 2 \pi\text{.}$ A way to plot this function in MATLAB would be the following sequence of commands:

>> f = @(x,y) x.*sin(x.*y)
>> [X,Y] = meshgrid(0:.1:5,pi:.01*pi:2*pi);
>> Z = f(X,Y)
>> mesh(X,Y,Z)

This will produce a 3-D plot that you can rotate by clicking on the rotate icon and then dragging with the mouse.Instead of the command mesh, you could use the command

>> surf(X,Y,Z)

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The key command in this sequence is [X Y] = meshgrid(a:h:b,c:k:d), which produces matrices of x and y values in X and Y. Enter:

>> size(X)
>> size(Y)
>> size(Z)

to see that each of these variables is a $101 \times 51$ matrix. To see the first few entries of X enter

>> X(1:6,1:6)

and to see the first few values of Y type

>> Y(1:6,1:6)

You should observe that the $x$ values in X begin at $0$ on the left column and increase from left to right. The $y$ values on the other have start at $\pi$ at the top and increase from top to bottom. Note that this arrangement is flipped from the usual arrangment in the $x$-$y$ plane.

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In the command [X Y] = meshgrid(a:h:b,c:k:d), $h$ is the increment in the $x$ direction and $k$ is the increment in the $y$ direction. Often we will calculate

\begin{equation*} h = \frac{b-a}{m}\quad \textrm{ and }\quad k = \frac{d-c}{n}\text{,} \end{equation*}

where $m$ is the number of intervals in the $x$ direction and $n$ is the number of intervals in the $y$ direction. To obtain a good plot it is best if $m$ and $n$ can be set between $10$ and $100\text{.}$

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A common way of visualizing a function of two variables is by a Contour Plot. In a contour plot we draw several level curves of the function, which are the curves at which the function is equal to a few values. A topographical map is an example of a contour plot. To produce a contour plot for the function $f(x,y)$ as above, since we have input the function itself and created a meshgrid on which we want to plot it, we simply input:

>> contour(X,Y,Z,10)

The optional number ``10’’ specify how many contour curves to display.

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For another example of meshgrid, contour and mesh, try the following and look at X and Y.

>> [X,Y] = meshgrid(0:.05:4,1:.02:2);
>> Z = (X+Y)./(1+X.^2+Y.^2);
>> contour(X,Y,Z,11)
>> mesh(X,Y,Z)

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Subsection 3.5.2 Scattered Data and Triangulation

Often we are interested in objects whose bases are not rectangular. For instance, data does not usually come arranged in a nice rectangular grid; rather, measurements are taken where convenient.

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In MATLAB we can produce triangles for a region by recording the coordinates of the vertices and recording which vertices belong to each triangle. The following script program produces such a set of triangles:

% mytriangles
% Program to produce a triangulation.
% V contains vertices, which are (x,y) pairs
V = [ 1/2 1/2  ;  1   1 ; 3/2 1/2 ;  .5  1.5 ;  0    0
      1    0  ;  2   0 ; 2    1  ; 1.5 1.5  ;  1    2
      0    2  ;  0    1]
% x, y are row vectors containing coordinates of vertices
x = V(:,1)';
y = V(:,2)';
%  Assign the triangles using Delaunay's algorithm
T = delaunay(x,y)

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You can plot the triangles using

>> trimesh(T,x,y)

You can also prescribe values (heights) at each vertex directly (say from a survey):

>> z1 = [ 2 3 2.5 2 1 1 .5 1.5 1.6 1.7 .9 .5 ];

or using a function:

>> f = @(x,y) abs(sin(x.*y)).^(3/2);
>> z2 = f(x,y);

The resulting profiles can be plotted:

>> trimesh(T,x,y,z1)
>> trisurf(T,x,y,z2)

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Each row of the matrix T corresponds to a triangle, so T(i,:) gives triangle number i. The three corners of triangle number i are at indices T(i,1), T(i,2), and T(i,3). So for example to get the $y$-coordinate of the second point of triangle number 5, enter

>> y(T(5,2))

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To see other examples of regions defined by triangles, download mywedge.m (Program A.2.9) and mywasher.m (Program A.2.8) and run them. Each of these programs defines vectors x and y of $x$ and $y$ values of vertices and a matrix T. As before T is a list of sets of three integers. Each triple of integers indicates which vertices to connect in a triangle.

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To plot a function, say $f(x,y) = x^{2} - y^{2}$ on the washer figure try

>> mywasher
>> z = x.^2 - y.^2
>> trisurf(T,x,y,z)

Note again that this plot can be rotated using the icon and mouse.

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Exercises 3.5.3 Exercises

1.

Plot the function $f(x,y) = \sin(x) \ e^{-x^2 - y^2}$ on the rectangle $-3 \le x \le 3\text{,}$ $-2 \le y \le 2$ using meshgrid and mesh. Make an appropriate choice of $m$ and $n$ and if necessary a rotation to produce a good plot. Calculate the $h$ and $k$ corresponding to your $m$ and $n\text{.}$ Turn in your plot and the calculation of $h$ and $k\text{.}$

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2.

Modeling after mywasher.m (Program A.2.8), produce

$A simple figure composed of 4 triangles.🔗$

using integer coordinates for the vertices. Use the axis command to zoom out so the outside edges are clearly visible. Compute $z = 3x +y^{2}$ and plot the graph. Turn in your program and the plots.

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