matlab中contourm,MATLAB 中contour函数的使用
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.
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