转自：http://msemac.redwoods.edu/~darnold/math50c/matlab/contours/index.xhtml

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).

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)=x2+y2.

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

x2+y2=c,

where c is

a constant. Of course, these "level curves" are circles, centered

at the origin, with radius c. These level

curves are drawn in Figure 2 for

constants c=0, 1, 2, 3, and

4.

Level curves of f(x,y)=x2+y2 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 meshgridcommand.

x=linspace(-3,3,40);

y=linspace(-3,3,40);

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

We then use the function f(x,y)=x2+y2, or

equivalently, z=x2+y2, 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 contourcommand 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 of f(x,y)=x2+y2 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 andh.

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.

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 provides a cleaner looking

plot.

Forcing Contours

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

Force more contours than the

default number provided by

the contour command.

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 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.

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 x2+2xy+y2-2x=3.

One way to proceed would be to first define the function

f(x,y)=x2+2xy+y2-2x,

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)=x2+2xy+y2-2x.

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.

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.

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.

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.

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 and

upload to your drop box.

Use contour to

sketch default level curves for the

function f(x,y)=1-x-y. Use

the clabelcommand to

automatically label the level curves.

Use contour to

sketch default level curves for the

function f(x,y)=xy. Use

the clabelcommand with the

'manual' switch to label level curves of choice.

Use contour to

sketch the level curves f(x,y)=c for f(x,y)=x2+4y2 for

the following values of c: 1,2,3,4, and

5.

Use

the contour command

to force 20 level curves for the

function f(x,y)=2+3x-2y.

Use

the meshc command

to produce a surface and contour plot for the

function (x,y)=9-x2-y2.

Use

the contour3 command

to sketch level curves at their heights for the

functionf(x,y)=x2+y2.

Use

the contour to

sketch the graph of the implicit

equation x3+y3=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.