Contour Maps in Matlab

In this activity we will introduce Matlab's contour command, which is used to plot the level curves of a multivariable function. Let's begin with a short discussion of the level curve concept.

Level Curves

Hikers and backpackers are likely to take along a copy of a topographical map when verturing into the wilderness (see Figure 1).

An image of a topo map.

A topographical map has lines of constant height.

If you walk along one of the contours shown in Figure 1, you will neither gain nor lose elevation. You're walking along a curve of constant elevation. If you walk directly perpendicular to a contour, then you are either walking directly downhill or uphill. When the contours are far apart, the gain or loss in elevation is gradual. When the contours are close together, the gain or loss in elevation is quite rapid.

The level curves of a multivariate function are analogous to the contours in the topographical map. They are curves of constant elevation. Let's look at an example.

Sketch several level curves of the function `f(x,y)=x^2+y^2`.

Solution: We are interested in finding points of constant elevation, that is, solutions of the equation

`f(x,y)=c`,

where `c` is a constant. Equivalently, we wish to sketch solutions of

`x^2+y^2=c`,

where `c` is a constant. Of course, these "level curves" are circles, centered at the origin, with radius `sqrt(c)`. These level curves are drawn in Figure 2 for constants `c=0`, 1, 2, 3, and 4`.

SVG

Level curves of `f(x,y)=x^2+y^2` lie in the `xy`-plane.

Matlab: It's a simple task to draw the level curves of Figure 2 using Matlab's contour command. We begin as if we were going to draw a surface, creating a grid of `(x,y)` pairs with the meshgrid command.

x=linspace(-3,3,40);
y=linspace(-3,3,40);
[x,y]=meshgrid(x,y);

We then use the function `f(x,y)=x^2+y^2`, or equivalently, `z=x^2+y^2`, to calculate the `z`-values.

z=x.^2+y.^2;

Where we would normally use the mesh command to draw the surface, instead we use the contour command to draw the level curves.

contour(x,y,z)

Add a grid, equalize, then tighten the axes.

grid on
axis equal
axis tight

Annotate the plot.

xlabel('x-axis')
ylabel('y-axis')
title('Level curves of the function f(x,y) = x^2 + y^2.')

The above sequence of commands will produce the level curves shown in Figure 3.

Level curves drawn with Matlab.

Level curves of `f(x,y)=x^2+y^2` drawn with Matlab's contour command.

By default, Matlab draws a few more level curves than the number shown in Figure 2.

Adding Labels to the Contours: It would be nice if we could label each contour with its height. As one might expect, Matlab has this capability. Using the same data as above, execute this command. Note that we use a semi-colon to suppress the output.

[c,h]=contour(x,y,z);

Without getting too technical, information on the level curves is stored in the output variables c and h. We then feed the output as input to Matlab's clabel command.

clabel(c,h)

Using the same formatting as above (grid, axis equal and tight, and annotations), this produces the image shown in Figure 4.

Label each contour with its height.

Label each contour with its height.

Adding Labels Manually: In Figure 4, there are labels all over the place, some that we might feel are not very well placed. We can exert control over how many labels are used and their placement. Simply pass the option 'manual' to Matlab's clabel command. First, redraw the contours, capturing again he output in the variables c and h.

[c,h]=contour(x,y,z);

Next, execute the clabel command with the 'manual' switch as follows.

clabel(c,h,'manual')

At first, it appears that nothing happens. However, move your mouse over the figure window and the axes and note that the mouse cursor turns into a large crosshairs. Each time you click a contour with the mouse, a label is set on the contour selected by the crosshairs. When you've completed clicking several contours, while the mouse crosshairs are still over the axes, press the Enter key on your keyboard. This will toggle the crosshairs off and stop further labeling of contours. You can now repeat the formatting (grid, equalize, tighten, and annoations) to produce the image in Figure 5.

Annotating level curves manually.

Annotating level curves manually provides a cleaner looking plot.

Forcing Contours

Sometimes you'd like to do one of two things:

  1. Force more contours than the default number provided by the contour command.
  2. Force contours at particular heights.

Forcing More Contours: You can force more contours by adding an additional argument to the contour command. To force 20 contours, execute the following command.

contour(x,y,z,20)

Adding the formatting commands (grid, equal and tighten, and annotations) produces the additional contours shown in Figure 6.

Forcing additional contours.

Forcing additional contours.

Forcing Specific Contours: You can also force contours at specific heights. To reproduce the level curves of Figure 1, at the heights `c=0`, 1, 2, 3, and 4, we pass the specific heights we wish to see in a vector to the contour command. First, list the specific heights in a vector.

v=[0,1,2,3,4];

Pass the vector v to the contour command as follows:

[c,h]=contour(x,y,z,v);

Labeling the contours shows that our contours have the heights requested.

clabel(c,h)

These commands, plus the formatting commands (grid, equalize and tighten, annotations) produce the result shown in Figure 7.

Forcing contours at particular heights.

Forcing contours at particular heights.

Note the strong resemblance of Figure 7 to Figure 1

Miscellaneous Extras

Implicit Plotting: Sometimes you want to draw a single contour. For example, suppose you wish to draw the graph of the implict relation `x^2+2xy+y2-2x=3`. One way to proceed would be to first define the function

`f(x,y)=x^2+2xy+y^2-2x`,

then plot the level curve `F(x,y)=3`. Start by creating a grid of `(,y)` pairs.

x=linspace(-3,3,40);
y=linspace(-3,3,40);
[x,y]=meshgrid(x,y);

Calculate `z=f(x,y)=x^2+2xy+y^2-2x`.

z=x.^2+2*x.*y+y.^2-2*x;

Now, we wish to draw the single contour `z=f(x,y)=3`. Create a vector with this height. Matlab requires that you repeat the height value you want two times.

v=[3,3];

Plot the single contour.

contour(x,y,z,v);

Add a grid, equalize and tighten the axes.

grid on
axis equal
axis tight

Finally, add appropriate annotations.

xlabel('x-axis')
ylabel('y-axis')
title('The implicit curve x^2+2xy+y^2-2x=3.')

The result of the above sequence of commands is captured in Figure 8.

Plotting an implicit equation.

Plotting an implicit equation.

Surface and Contours: Sometimes you want the surface and the contours. Again, an easy task in Matlab. The following commands produce the surface and contour plot shown in Figure 9.

x=linspace(-3,3,40);
y=linspace(-3,3,40);
[x,y]=meshgrid(x,y);
z=x.^2+y.^2;
meshc(x,y,z);
grid on
box on
view([130,30])
xlabel('x-axis')
ylabel('y-axis')
zlabel('z-axis')
title('Mesh and contours for f(x,y)=x^2+y^2.')

Note that the meshc command provides both a mesh and a contour plot.

Surface and contours combined.

Surface and contours combined.

In Figure 9, note that when the level curves in the plane get close together, the corresponding position on the surface is steeper. On the other hand, when the distance between the level curves is large, the surface is flatter in nature; i.e., the elevation change is gradual.

Contours Plotted at Actual Height: Finally, it's also possible to plot the contours at their actual heights.

x=linspace(-3,3,40);
y=linspace(-3,3,40);
[x,y]=meshgrid(x,y);
z=x.^2+y.^2;
contour3(x,y,z);
grid on
box on
view([130,30])
xlabel('x-axis')
ylabel('y-axis')
zlabel('z-axis')
title('Contours at height for f(x,y)=x^2+y^2.')

In Figure 10, note that the contour3 command plots contours at their actual heights instead of in the plane. This hands us a deeper understanding of the meaning of a "level curve."

Contours plotted at actual heights.

Contours plotted at actual heights.

Matlab Files

Although the following file features advanced use of Matlab, we include it here for those interested in discovering how we generated the images for this activity. You can download the Matlab file at the following link. Download the file to a directory or folder on your system.

level.m

The file level.m is designed to be run in "cell mode." Open the file level.m in the Matlab editor, then enable cell mode from the Cell Menu. After that, use the entries on the Cell Menu or the icons on the toolbar to execute the code in the cells provided in the file. There are options for executing both single and multiple cells. After executing a cell, examine the contents of your folder and note that a PNG file was generated by executing the cell.

Exercises

When completed, publish the results of these exercises to HTML print the results. Hand in the printed results in class.

  1. Use contour to sketch default level curves for the function `f(x,y)=sqrt(1-x-y)`. Use the clabel command to automatically label the level curves.
  2. Use contour to sketch default level curves for the function `f(x,y)=xy`. Use the clabel command with the 'manual' switch to label level curves of choice.
  3. Use contour to sketch the level curves `f(x,y)=c` for `f(x,y)=x^2+4y^2` for the following values of `c`: 1,2,3,4, and 5.
  4. Use the contour command to force 20 level curves for the function `f(x,y)=2+3x-2y`.
  5. Use the meshc command to produce a surface and contour plot for the function `(x,y)=9-x^2-y^2`.
  6. Use the contour3 command to sketch level curves at their heights for the function `f(x,y)=sqrt(x^2+y^2)`.
  7. Use the contour to sketch the graph of the implicit equation `x^3+y^3=3xy`. This curve is known as the Folium of Descartes. Note: You are asked to plot a single cuver here, not a set of many contours.