Matlab Chapter6.pdf

Subjects:: Graphics
lecturer in charge: Dr. Muzafar F. HAMA
Tel: 0771 015 4795
EMail: hamamuzafar@yahoo.com
date: 3-2-2011
Subject objective:
MATLAB has powerful and versatile graphics capabilities. Figures of many types can be
generated with relative ease and their“look and feel” is highly customizable. In this chapter we
cover the basic use of MATLAB’s most popular tools for graphing two- and three-dimensional
data.
1 Basic 2-D Graphs
Graphs (in 2-D) are drawn with the plot statement.The plot function has different forms,
depending on the input arguments. If y is a vector, plot(y) produces a piecewise linear graph
of the elements of y versus the index of the elements of y. If you specify two vectors as
arguments, plot(x, y) produces a graph of y versus x. The plot command also accepts matrix
arguments. For example, these statements use the colon operator to create a vector of x values
ranging from 0 to 2π, compute the sine of these values, and plot the result:
≫ x = 0 : pi/100 : 2 ∗ pi;
≫ y = sin(x);
≫ plot(x, y)
If x is an m-vector and y is an m-by-n matrix, plot(x, y) superimposes the plots created by
x and each column of y. Similarly, if x and y are both m-by-n, plot(x, y) superimposes the
plots created by corresponding columns of x and y.
Straight-line graphs are drawn by giving the x and y coordinates of the end-points in two
vectors. For example, to draw a line between the point with cartesian coordinates (0, 1) and
(4, 3) use the statement
≫ plot([0 4], [1 3])
i.e. [0 4] contains the x coordinates of the two points, and [1 3] contains their y coordinates.
1.1 Plotting Multiple Data Sets in One Graph
Multiple x-y pair arguments create multiple graphs with a single call to plot. MATLAB automatically
cycles through a predefined (but user settable) list of colors to allow discrimination
among sets of data.For example, these statements plot three related functions of x, with each
curve in a separate distinguishing color:
≫ x = 0 : pi/100 : 2 ∗ pi;
≫ y = sin(x);
≫ y2 = sin(x − .25);
≫ y3 = sin(x − .5);
≫ plot(x, y, x, y2, x, y3)
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1.2 Specifying Line Styles and Colors
It is possible to specify color, line styles, and markers (such as plus signs or circles) when you
plot your data using the plot command:
≫ plot(x, y, ′ color style marker ′ )
color style marker is a string containing from one to four characters (enclosed in single
quotation marks) constructed from a color, a line style, and a marker type:
• Color strings are ′ c ′ , ′ m ′ , ′ y ′ , ′ r ′ , ′ g ′ , ′ b ′ , ′ w ′ , and ′ k ′ . These correspond to cyan, magenta,
yellow, red, green, blue, white, and black.
• Line style strings are ′ − ′ for solid, ′ −− ′ for dashed, ′ : ′ for dotted, and ′ −. ′ for dash-dot.
Omit the line style for no line.
• The marker types are ′ + ′ , ′ ◦ ′ , ′ ∗ ′ , and ′ x ′ , and the filled marker types are ′ s ′ for square,
′ d ′ for diamond, ′ ∧ ′ for up triangle, ′ v ′ for down triangle, ′ > ′ for right triangle, ′ < ′ for
left triangle, ′ p ′ for pentagram, ′ h ′ for hexagram, and none for no marker.
Example 1.1
≫ x = 0 : pi/100 : 2 ∗ pi;
≫ y = sin(x);
≫ plot(x, y, ′ ks ′ )
plots black squares at each data point, but does not connect the markers with a line.
Example 1.2 The statement
≫ x = 0 : pi/100 : 2 ∗ pi;
≫ y = sin(x);
≫ plot(x, y, ′ r : + ′ )
plots a red dotted line and places plus sign markers at each data point.
Example 1.3 This example plots the data twice using a different number of points for the
dotted line and marker plots:
x1 = 0 : pi/100 : 2 ∗ pi;
x2 = 0 : pi/10 : 2 ∗ pi;
plot(x1, sin(x1), ′ r : ′ , x2, sin(x2), ′ r+ ′ )
1.3 Graphing Imaginary and Complex Data
When the arguments to plot are complex, the imaginary part is ignored except when you
pass plot a single complex argument. For this special case, the command is a shortcut for a
graph of the real part versus the imaginary part. Therefore,
plot(Z)
where Z is a complex vector or matrix, is equivalent to plot(real(Z), imag(Z)) For example,
≫ t = 0 : pi/10 : 2 ∗ pi;
≫ plot(exp(i ∗ t), ′ −o ′ )
draws a 20-sided polygon with little circles at the vertices.
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1.4 Adding Plots to an Existing Graph
The hold command enables you to add plots to an existing graph. When you type
≫ holdon
MATLAB does not replace the existing graph when you issue another plotting command;
it adds the new data to the current graph, rescaling the axes if necessary.
For example, these statements first create a contour plot of the peaks function, then superimpose
a pseudocolor plot of the same function:
≫ x = 0 : pi/100 : 2 ∗ pi;
≫ y = sin(x);
≫ plot(x, y)
≫ holdon
≫ plot(x, cos(x))
≫ holdoff
For More Information See “Creating Specialized Plots” in [1].
1.5 Displaying Multiple Plots in One Figure
The subplot command enables you to display multiple plots in the same window or print them
on the same piece of paper. Typing
≫ subplot(m, n, p)
partitions the figure window into an m-by-n matrix of small subplots and selects the p−th
subplot for the current plot. The plots are numbered along the first row of the figure window,
then the second row, and so on. For example, subplot(425) splits the figure window into a
4-by-2 matrix of regions and specifies that plotting commands apply to the fifth region.For
example, these statements plot data in four different subregions of the figure window:
≫ t = 0 : pi/10 : 2 ∗ pi;
≫ [X, Y, Z] = cylinder(4 ∗ cos(t));
≫ subplot(2, 2, 1); mesh(X)
≫ subplot(2, 2, 2); mesh(Y )
≫ subplot(2, 2, 3); mesh(Z)
≫ subplot(2, 2, 4); mesh(X, Y, Z)
Remark 1.1 It is possible to produce irregular grids of plots by invoking subplot with different
grid patterns. For example,:
x = linspace(0, 15, 100);
subplot(2, 2, 1), plot(x, sin(x))
subplot(2, 2, 2), plot(x, round(x))
subplot(2, 1, 2), plot(x, sin(round(x)))
The third argument to subplot can be a vector specifying several regions, so we could replace
the last line by
subplot(2, 2, 3 : 4), plot(x, sin(round(x)))
1.6 Controlling the Axes
The axis command provides a number of options for setting the scaling, orientation, and aspect
ratio of graphs.
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1.6.1 Setting Axis Limits
By default, MATLAB finds the maxima and minima of the data and chooses the axis limits to
span this range. The axis command enables you to specify your own limits:
≫ axis([xmin xmax ymin ymax])
or for three-dimensional graphs,
≫ axis([xmin xmax ymin ymax zmin zmax])
Use the command
≫ axis auto
to reenable MATLAB automatic limit selection.
1.6.2 Setting the Axis Aspect Ratio
The axis command also enables you to specify a number of predefined modes. For example,
≫ axis square
makes the x-axis and y-axis the same length.
≫ axis equal
makes the individual tick mark increments on the x-axes and y-axes the same length. This
means
≫ plot(exp(i ∗ [0 : pi/10 : 2 ∗ pi]))
followed by either axis square or axis equal turns the oval into a proper circle:
≫ axis auto normal
returns the axis scaling to its default automatic mode.
1.6.3 Setting Axis Visibility
You can use the axis command to make the axis visible or invisible.
≫ axis on
makes the axes visible. This is the default.
≫ axis off
makes the axes invisible.
1.6.4 Setting Grid Lines
The grid command toggles grid lines on and off. The statement
≫ grid on
turns the grid lines on, and
≫ grid off
turns them back off again. For More Information See the axis and axes reference pages and
“Axes Properties” in [1].
1.7 Adding Axis Labels and Titles
The xlabel( ), ylabel( ), and zlabel( ) commands add x-, y-, and z-axis labels. The title(text)
command adds a text at the top of the figure and the text function inserts text anywhere in
the figure. You can produce mathematical symbols using LaTeX notation in the text string, as
the following example illustrates:
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Example 1.4 The characters \ pi create the symbol π.
Example 1.5
≫ xlabel( ′ x = 0 : 2\pi ′ )
≫ ylabel( ′ Sine of x ′ )
≫ title( ′ Plot of the Sine Function’,’FontSize ′ , 12)
≫ t = −pi : pi/100 : pi;
≫ y = sin(t);
≫ plot(t, y)
≫ axis([−pi pi − 1 1])
≫ xlabel( ′ −\pi ≤ t ≤ \pi ′ )
≫ ylabel( ′ sin(t) ′ )
≫ title(’Graph of the sine function’)
≫ text(1, −1/3, ′ Notetheoddsymmetry. ′ )
You can also set these options interactively. Note that the location of the text string is defined
in axes units (i.e., the same units as the data).
Remark 1.2 Graphs may be labeled with the following statements:
≫ gtext(text)
writes a string (text) in the graph window. gtext puts a cross-hair in the graph window
and waits for a mouse button or keyboard key to be pressed. The cross-hair can be positioned
with the mouse or the arrow keys. For example,
≫ gtext(Xmarksthespot)
Text may also be placed on a graph interactively with Tools click Edit Plot from the figure
window.
≫ text(x, y, text)
writes text in the graphics window at the point specified by x and y.
If x and y are vectors, the text is written at each point. If the text is an indexed list,
successive points are labeled with corresponding rows of the text.
≫ title(text)
1.8 Figure Windows
Graphing functions automatically open a new figure window if there are no figure windows
already on the screen. If a figure window exists, MATLAB uses that window for graphics
output. If there are multiple figure windows open, MATLAB targets the one that is designated
the “current figure” (the last figure used or clicked in).
To make an existing figure window the current figure, you can click the mouse while the
pointer is in that window or you can type
≫ figure(n)
where n is the number in the figure title bar. The results of subsequent graphics commands
are displayed in this window.
≫ x = 0 : pi/100 : 2 ∗ pi;
≫ y = sin(x);
≫ y2 = sin(x − .25);
≫ y3 = sin(x − .5);
≫ plot(x, y)
≫ plot(x, y2)
≫ plot(x, y3)
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To open the second figure window and make it the current figure, type
≫ figure(2)
To open a new figure window and make it the current figure, type
≫ figure
Clearing the Figure for a New Plot When a figure already exists, most plotting commands
clear the axes and use this figure to create the new plot. However, these commands do not
reset figure properties, such as the background color or the colormap. If you have set any figure
properties in the previous plot, you might want to use the clf command with the reset option,
≫ clf reset
before creating your new plot to restore the figures properties to their defaults.
clf clears the current figure window. It also resets all properties associated with the axes,
such as the hold state and the axis state.
cla deletes all plots and text from the current axes, i.e. leaves only the x- and y-axes and
their associated information.
For More Information See “Figure Properties” and “Graphics Windows” the Figure in [1].
1.9 Plotting rapidly changing mathematical functions: fplot
In all the graphing examples so far, the x coordinates of the points plotted have been incremented
uniformly, e.g. x = 0 : 0.01 : 4. If the function being plotted changes very rapidly in
some places, this can be inefficient, and can even give a misleading graph.
For example, the statements
≫ x = 0.01 : .001 : .1;
≫ plot(x, sin(1./x))
But if the x increments are reduced to 0.0001, we get another graph instead. For x < 0.04, the
two graphs look quite different.
MATLAB has a function called fplot which uses a more elegant approach. Whereas the
above method evaluates sin(1/x) at equally spaced intervals, fplot evaluates it more frequently
over regions where it changes more rapidly. Here’s how to use it:
≫ fplot( sin(1/x), [0.01 0.1])% no, 1./x not needed!
The general pattern is fplot(fun, lims, tol, N, ′ LineSpec ′ , pl, p2, . . .). The argument list
works as follows.
• fun specifies the function to be plotted.
• The x and/or y limits are given by lims.
• tol is a relative error tolerance, the default value of 2 × 10 −3 corresponding to 0.2%
accuracy.
• At least N + 1 points will be used to produce the plot.
• LineSpec determines the line type.
• p1, p2, . . . are parameters that are passed to fun, which must have input arguments x,
p1, p2, . . . .
The arguments tol, N, and ’LineSpec’ can be specified in any order, and an empty matrix
([ ]) can be passed to obtain the default for any of these arguments.
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Example 1.6 In this example, fplot produces a graph of the function exp( √ x sin 12x) over the
interval 0 ≤ x ≤ 2π. In the second call, we override the default solid line style and specify a
dashed line with ′ − − ′ . The argument [0.01 1 − 15 20] in the third call forces limits in both the
x and y directions, 0.01 ≤ x ≤ 1 and −15 ≤ y ≤ 20, and ′ − . ′ asks for a dash-dot line style.
The final fplot example illustrates how more than one function can be plotted in the same call.
subplot(221), fplot( ′ exp(sqrt(x) ∗ sin(12 ∗ x)) ′ , [0 2 ∗ pi])
subplot(222), fplot( ′ sin(round(x)) ′ , [0 10], ′ −− ′ )
subplot(223), fplot( ′ cos(30 ∗ x)/x ′ , [0.01 1 − 15 20], ′ −. ′ )
subplot(224), fplot( ′ [sin(x), cos(2 ∗ x), 1/(1 + x)] ′ , [05 ∗ pi − 1.5 1.5])
Remark 1.3 We remark that:
• The two statements plot(x, Y, ′ ms − − ′ ) and plot(x, Y, ′ s − −m ′ ) are equivalent.
• You can exert further control by supplying more arguments to plot. The properties
LineWidth (default 0.5 points) and MarkerSize (default 6 points) can be specified in
points, where a point is 1/72 inch. For example, the commands
≫ plot(x, y, ′ LineWidth ′ , 2)
≫ plot(x, y, ′ p ′ , ′ MarkerSize ′ , 10)
produce a plot with a 2-point line width and 10-point marker size, respectively. For markers
that have a well-defined interior, the MarkerEdgeColor and MarkerFaceColor can
be set to one of the colors in subsection 1.2. So, for example,
≫ plot(x, y, ′ o ′ , ′ Marker Edge Color ′ , ′ m ′ )
gives magenta edges to the circles.
≫ plot(x, y, ′ m − −− ′ , ′ LineWidth ′ , 3, ′ MarkerSize ′ , 5)
and the right-hand plot with
≫ plot(x, y, ′ − − rs ′ , ′ MarkerSize ′ , 20, ′ MarkerFaceColor ′ , ′ g ′ )
Default values for these properties, and for some others to be discussed later in the chapter,
are summarized in Table 6.1.
centerlineTable 6.1. Default values for some properties.
LineWidth 0.5
MarkerSize 6
MarkerEdgeColor auto
MarkerFaceColor none
FontSize 10
FontAngle normal
• Using loglog instead of plot causes the axes to be scaled logarithmically. This feature is
useful for revealing power-law relationships as straight lines. In the example below we plot
|1 + h + h 2 /2 − exp(h)| against h for h = 1, 10 −1 , 10 −2 , 10 −3 , 10 −4 . This quantity behaves
like a multiple of h 3 when h is small, and hence on a log-log scale the values should lie
close to a straight line of slope 3. To confirm this, we also plot a dashed reference line
with the predicted slope, exploiting the fact that more than one set of data can be passed
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to the plot commands.
h = 10. ∧ [0 : −1 : −4];
≫ taylerr = abs(1 + h + h. ∧ 2/2) − exp(h));
≫ loglog(h, taylerr, ′ − ′ , h, h. ∧ 3, ′ −− ′ )
≫ xlabel( ′ h ′ )
≫ ylabel( ′ Absolute value ′ , ′ of error ′ )
≫ title( ′ Error in quadratic Taylor series approximation to exp(h) ′ )
≫ box off
In this example, we used title, xlabel, and ylabel. These functions reproduce their input
string above the plot and on the x- and y-axes, respectively. We also used the command
box off, which removes the box from the current plot, leaving just the x- and y-axes.
MATLAB will, of course, complain if nonpositive data is sent to loglog (it displays a
warning and plots only the positive data).
Table 6.2. Some commands for controlling the axes.
axis([xmin xmax ymin ymax])
axis auto
axis equal
axis off
axis square
axis tight
xlim([xmin xmax])
ylim([ymin ymax])
Set specified x- and y-axis limits
Return to default axis limits
Equalize data units on x-, y-, and z-axes
Remove axes
Make axis box square (cubic)
Set axis limits to range of data
Set specified x-axis limits
Set specified y-axis limits
The command close closes the current figure window. The command close all causes all
the figure windows to be closed.
Example 1.7 [2] The following program:
• Plots the function 1/(x − 1) 2 + 3/(x − 2) 2 over the interval [0, 3]:
x = linspace(0, 3, 500);
plot(x, 1./(x − 1). ∧ 2 + 3./(x − 2). ∧ 2)
grid on
We specified grid on, which introduces a light horizontal and vertical hashing that extends
from the axis ticks. Because of the singularities at x = 1, 2 the plot is uninformative.
However, by executing the additional command
ylim([0 50])
which focuses on the interesting part of the first plot.
• Plot the epicycloid
0 ≤ t ≤ 10π
{ x(t) = (a + b) cos(t) − b cos((a/b + l)t))
y(t) = (a + b) sin(t) − b sin((a/b + l)t)
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for a = 12 and b = 5.
a = 12; b = 5;
t = 0 : 0.05 : 10 ∗ pi;
x = (a + b) ∗ cos(t) − b ∗ cos(a/b + l) ∗ t);
y = (a + b) ∗ sin(t) − b ∗ sin(a/b + l) ∗ t);
plot(x, y)
axis equal
axis([−25 25 − 25 25])
grid on
title( ′ Epicycloid: a = 12, b = 5 ′ )
xlabel( ′ x(t) ′ ), ylabel( ′ y(t) ′ )
• Plot the Legendre polynomials of degrees 1 to 4 (for the properties of these polynomials,
see, help of matlab) and use the legend function to add a box that explains the line styles.
x = −1 : .01 : 1;
p1 = x;
p2 = (3/2) ∗ x. ∧ 2 − 1/2;
p3 = (5/2) ∗ x. ∧ 3 − (3/2) ∗ x;
p4 = (35/8) ∗ x. ∧ 4 − (15/4) ∗ x. ∧ 2 + 3/8;
plot(x, pl, ′ r : ′ , x, p2, ′ g − − ′ , x, p3, ′ b − . ′ , x, p4, ′ m− ′ )
box off
legend( ′ \it n = 1 ′ , ′ \it n = 2 ′ , ′ \it n = 3 ′ , ′ \it n = 4 ′ , ′ Location ′ , ′ SouthEast ′ )
xlabel( ′ x ′ , ′ FontSize ′ , 12, ′ FontAngle ′ , ′ italic ′ )
ylabel( ′ P n ′ , ′ FontSize ′ , 12, ′ FontAngle ′ , ′ italic ′ , ′ Rotation ′ , 0)
title( ′ Legendre Polynomials ′ , ′ FontSize ′ , 14)
text(−.6, .7, ′ (n + 1)P n + 1(x) = (2n + 1)xP n(x) − nP n − 1(x) ′ , . . .
′ FontSize ′ , 12, ′ FontAngle ′ , ′ italic ′ )
Remark 1.4 Generally, typing legend(’string ′ 1, ’string 2 ’,. . ., ’string n ’) will create a legend
box that puts ’string i ’ next to the color/marker/line style information for the corresponding
plot. By default, the box appears in the top right-hand (northeast) corner of the axis area. The
location of the box can be specified with the syntax legend(’,. . ., ’Location’, location), where
location is a string with possible values that include:
’North’
’NorthWest’
’NorthOutside’
’Best’
’BestOutside’
inside plot box near top
inside top left
outside plot box near top
automatically chosen to give least conflict with data
automatically chosen to leave least unused space outside plot
These values can be abbreviated as ’N’, ’NW ’, etc. In our example we chose the bottom righthand
corner. Once the plot has been drawn, the legend box can be repositioned by putting the
cursor over it and dragging it using the left mouse button. The legend function has many other
options, which can be seen by typing doc legend.
2 3-D plots
MATLAB has a variety of functions for displaying and visualizing data in 3-D, either as lines
in 3-D, or as various types of surfaces. This section provides a brief overview.
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2.1 plot3
The function plot3 is the 3-D version of plot. The command
plot3(x, y, z)
draws a 2-D projection of a line in 3-D through the points whose coordinates are the elements
of the vectors x, y and z. The following example illustrates the simplest usage: plot3(x, y, z)
draws a “join-the-dots” curve by taking the points x(i), y(i), z(i) in order.
t = −5 : .005 : 5;
x = (1 + t. − 2). ∗ sin(20 ∗ t);
y = (1 + t. − 2). ∗ cos(20 ∗ t);
z = t;
plot3(x, y, z)
grid on
F S = ′ FontSize ′ ;
xlabel( ′ x(t) ′ , F S, 14), ylabel( ′ y(t) ′ , F S, 14)
zlabel( ′ z(t) ′ , F S, 14, ′ Rotation ′ , 0)
title( ′ plot3example ′ , FS, 14 )
This example also uses the functions xlabel, ylabel, and title, which were discussed in the
previous section, and the analogous zlabel. The color, marker, and line styles for plot3 can
be controlled in the same way as for plot. So, for example, plot3(x, y, z, ′ rx − − ′ ) would use
a red dashed line and place a cross at each point. Note that for 3D plots the default is box
off; specifying box on adds a box that bounds the plot.
For example, the command
plot3(rand(1, 10), rand(1, 10), rand(1, 10))
generates 10 random points in 3-D space, and joins them with lines.
As another example, the statements
≫ t = 0 : pi/50 : 10 ∗ pi;
≫ plot3(exp(−0.02 ∗ t). ∗ sin(t), exp(−0.02 ∗ t). ∗ cos(t), ...
xlabel(x-axis), ylabel(y-axis), zlabel(z-axis))
2.2 Mesh and Surface Plots
MATLAB defines a surface by the z-coordinates of points above a grid in the x-y plane, using
straight lines to connect adjacent points. The mesh and surf plotting functions display surfaces
in three dimensions. mesh produces wireframe surfaces that color only the lines connecting the
defining points. surf displays both the connecting lines and the faces of the surface in color.
Example 2.1 (Mesh surfaces)
[x y] = meshgrid(−8 : 0.5 : 8);
r = sqrt(x. ∧ 2 + y. ∧ 2) + eps;
z = sin(r)./r;
Remark 2.1 The mesh function can also be used to ‘Visualize’ a matrix.
≫ a = zeros(30, 30);
≫ a(:, 15) = 0.2 ∗ ones(30, 1);
≫ a(7, :) = 0.1 ∗ ones(1, 30);
≫ a(15, 15) = 1;
≫ mesh(a)
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The matrix a is 30 × 30. The element in the middle a(15, 15) is 1, all the elements in row 7
are 0.1, and all the remaining elements in column 15 are 0.2. mesh(a) then interprets the rows
and columns of a as an x-y coordinate grid, with the values a(i, j) forming the mesh surface
above the points (i, j).
2.2.1 Contour plots
If you managed to draw a plot try the command
≫ contour(u)
You should get a contour plot of the heat distribution for example. Here’s the code:
≫ [x, y] = meshgrid(−2.1 : 0.15 : 2.1, −6 : 0.15 : 6);
≫ u = 80 ∗ y. ∧ 2. ∗ exp(−x. ∧ 2 − 0.3 ∗ y. ∧ 2);
≫ contour(u)
Remark 2.2 The function contour can take a second input variable. It can be a scalar specifying
how many contour levels to plot, or it can be a vector specifying the values at which to
plot the contour levels.
Remark 2.3 A simple contour plotting facility is provided by ezcontour. ezcontour in the
following example produces contours for the function sin(3y − x 2 + 1) + cos(2y 2 − 2x) over the
range −2 ≤ x ≤ 2 and −1 ≤ y ≤ 1;
subplot(211)
ezcontour( ′ sin(3 ∗ y − x ∧ 2 + 1) + cos(2 ∗ y ∧ 2 − 2 ∗ x) ′ , [−2 2 − 1 1]);
x = −2 : .01 : 2; y = −1 : .01 : 1;
[X, Y]=meshgrid(x, y);
Z = sin(3 ∗ Y − X. ∧ 2 + 1) + cos(2 ∗ Y. ∧ 2 − 2 ∗ X);
subplot(212)
contour(x, y, Z, 20)
Example 2.2 (meshc) A 3-D contour may be drawn under a surface as follows:
≫ [x y] = meshgrid(−2 : .2 : 2);
≫ z = x. ∗ exp(−x. ∧ 2 − y. ∧ 2);
≫ meshc(z)
Example 2.3 (Cropping a surface with NaNs) If a matrix for a surface plot contains a
data of type NaNs, these elements are not plotted. This enables you to cut away (crop) parts
of a surface.
[x y] = meshgrid(−2 : .2 : 2, −2 : .2 : 2);
z = x. ∗ exp(−x. ∧ 2 − y. ∧ 2);
c = z; % preserve the original surface
c(1 : 11, 1 : 21) = nan ∗ c(1 : 11, 1 : 21);
mesh(c), xlabel(x-axis), ylabel(y-axis)
Example 2.4 Consider the scalar function of two variables V = x 2 + y. The gradient of V is
defined as the vector field
( ∂V
∇V =
∂x , ∂V )
= (2x, 1).
∂y
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The following statements draw arrows indicating the direction of ∇V at points in the x − y
plane:
≫ [x y] = meshgrid(−2 : .2 : 2, −2 : .2 : 2);
≫ V = x. ∧ 2 + y;
≫ dx = 2 ∗ x;
≫ dy = dx; % dy same size as dx
≫ dy(:, :) = 1; % now dy is same size as dx but all 1’s
≫ contour(x, y, V )
The ‘contour’ lines indicate families of level surfaces; the gradient at any point is perpendicular
to the level surface which passes through that point. The vectors x and y are needed in the call
to contour to specify the axes for the contour plot.
Example 2.5 (Rotation of 3-D graphs) The view function enables you to specify the angle
from which you view a 3-D graph. To see it in operation, run the following program:
2.2.2 Polar plots
≫ a = zeros(30, 30);
≫ a(:, 15) = 0.2 ∗ ones(30, 1);
≫ a(7, :) = 0.1 ∗ ones(1, 30);
≫ a(15, 15) = 1;
≫ mesh(a)
≫ for az = −37.5 : 15 : −37.5 + 360
≫ mesh(a), view(az, el)
≫ pause(0.5)
≫ end
The point (x, y) in cartesian coordinates is represented by the point (θ, r) in polar coordinates,
where
x = r cos(θ),
y = r sin(θ),
and θ varies between 0 and 2θ radians (360 ◦ ).
The command polar(θ, r) generates a polar plot of the points with angles in theta and
magnitudes in r.
As an example, the statements
References
≫ x = 0 : pi/40 : 2 ∗ pi;
≫ polar(x, sin(2 ∗ x)), grid
[1] –, “Matlab7 Graphics”, The MathWorks, 2007.
[2] –, “Matlab7 Programming”, The MathWorks, 2007.
[3] –, “Matlab7 3-D Visualization”, The MathWorks, 2007.
[4] B. Hahn and D. T. Valentine, “Essential Matlab For Engineers and Scientists”, Elsevier,
third edition, 2007.
[5] D. J. Higham and N. J. Higham, “Matlab Guide”, SIAM puplications, 2000.
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