Chapter 5 MATLAB Graphics In this chapter we describe more of MATLAB’s graphics commands and the most common ways of manipulating and customizing them. You can get a list of MATLAB graphics commands by typing help graphics (for general graphics commands), help graph2d (for two-dimensional graphing), help graph3d (for three-dimensional graphing), or help specgraph (for special- ized graphing commands). We have already discussed the commands plot and ezplot in Chapter 2. We will begin this chapter by discussing more uses of these commands, as well as the other most commonly used plotting commands in two and three dimen- sions. Then we will discuss some techniques for customizing and manipulating graphics. Two-Dimensional Plots Often one wants to draw a curve in the x-y plane, but with y not given explicitly as a function of x. There are two main techniques for plotting such curves: parametric plotting and contour or implicit plotting. We discuss these in turn in the next two subsections. ParametricPlots Sometimes x and y are bothgiven as functions of some parameter. For example, the circle of radius 1 centered at (0,0) can be expressed in parametric form as x = cos(2πt), y = sin(2πt) where t runs from 0 to 1. Though y is not expressed as a function of x, you can easily graphthis curve with plot, as follows: >> T = 0:0.01:1; 67 68 Chapter 5: MATLAB Graphics >> plot(cos(2*pi*T), sin(2*pi*T)) >> axis square -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Figure 5-1 The output is shown in Figure 5.1. If you had used an increment of only 0.1in the T vector, the result would have been a polygon with clearly visible corners, an indication that you should repeat the process with a smaller increment until you get a graph that looks smooth. If you have version 2.1 or higher of the Symbolic Math Toolbox (cor- responding to MATLAB version 5.3 or higher), then parametric plotting is also possible with ezplot. Thus one can obtain almost the same picture as Figure 5-1 withthe command >> ezplot(’cos(t)’, ’sin(t)’, [0 2*pi]); axis square Two-Dimensional Plots 69 Contour Plots and Implicit Plots A contour plot of a function of two variables is a plot of the level curves of the function, that is, sets of points in the x-y plane where the function assumes a constant value. For example, the level curves of x 2 + y 2 are circles centered at the origin, and the levels are the squares of the radii of the circles. Contour plots are produced in MATLAB with meshgrid and contour. The command meshgrid produces a grid of points in a specified rectangular region, witha specified spacing. This grid is used by contour to produce a contour plot in the specified region. We can make a contour plot of x 2 + y 2 as follows: >> [X Y] = meshgrid(-3:0.1:3, -3:0.1:3); >> contour(X, Y, X.ˆ2 + Y.ˆ2) >> axis square The plot is shown in Figure 5-2. We have used MATLAB’s vector notation to -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 Figure 5-2 70 Chapter 5: MATLAB Graphics produce a grid withspacing 0.1 in bothdirections. We have also used axis square to force the same scale on both axes. You can specify particular level sets by including an additional vector ar- gument to contour. For example, to plot the circles of radii 1, √ 2, and √ 3, type >> contour(X, Y, X.ˆ2 + Y.ˆ2, [1 2 3]) The vector argument must contain at least two elements, so if you want to plot a single level set, you must specify the same level twice. This is quite useful for implicit plotting of a curve given by an equation in x and y.For example, to plot the circle of radius 1 about the origin, type >> contour(X, Y, X.ˆ2 + Y.ˆ2, [1 1]) Or to plot the lemniscate x 2 − y 2 = (x 2 + y 2 ) 2 , rewrite the equation as (x 2 + y 2 ) 2 − x 2 + y 2 = 0 and type >> [X Y] = meshgrid(-1.1:0.01:1.1, -1.1:0.01:1.1); >> contour(X, Y, (X.ˆ2 + Y.ˆ2).ˆ2 - X.ˆ2 + Y.ˆ2, [0 0]) >> axis square >> title(’The lemniscate xˆ2-yˆ2=(xˆ2+yˆ2)ˆ2’) The command title labels the plot with the indicated string. (In the default string interpreter, ˆ is used for inserting an exponent and is used for sub- scripts.) The result is shown in Figure 5-3. If you have the Symbolic Math Toolbox, contour plotting can also be done withthe command ezcontour, and implicit plotting of a curve f (x, y)= 0 can also be done with ezplot. One can obtain almost the same picture as Figure 5-2 withthe command >> ezcontour(’xˆ2 + yˆ2’, [-3, 3], [-3, 3]); axis square and almost the same picture as Figure 5-3 with the command >> ezplot(’(xˆ2 + yˆ2)ˆ2 - xˆ2 + yˆ2’, . [-1.1, 1.1], [-1.1, 1.1]); axis square Two-Dimensional Plots 71 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 The lemniscate x 2 -y 2 =(x 2 +y 2 ) 2 Figure 5-3 Field Plots The MATLAB routine quiver is used to plot vector fields or arrays of arrows. The arrows can be located at equally spaced points in the plane (if x and y coordinates are not given explicitly), or they can be placed at specified loca- tions. Sometimes some fiddling is required to scale the arrows so that they don’t come out looking too big or too small. For this purpose, quiver takes an optional scale factor argument. The following code, for example, plots a vector field witha “saddle point,” corresponding to a combination of an attractive force pointing toward the x axis and a repulsive force pointing away from the y axis: >> [x, y] = meshgrid(-1.1:.2:1.1, -1.1:.2:1.1); >> quiver(x, -y); axis equal; axis off The output is shown in Figure 5-4. 72 Chapter 5: MATLAB Graphics Figure 5-4 Three-Dimensional Plots MATLAB has several routines for producing three-dimensional plots. Curves in Three-Dimensional Space For plotting curves in 3-space, the basic command is plot3, and it works like plot, except that it takes three vectors instead of two, one for the x coordi- nates, one for the y coordinates, and one for the z coordinates. For example, we can plot a helix (see Figure 5-5) with >> T = -2:0.01:2; >> plot3(cos(2*pi*T), sin(2*pi*T), T) Again, if you have the Symbolic Math Toolbox, there is a shortcut using ezplot3; you can instead plot the helix with >> ezplot3(’cos(2*pi*t)’, ’sin(2*pi*t)’, ’t’, [-2, 2]) Three-Dimensional Plots 73 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Figure 5-5 Surfaces in Three-Dimensional Space There are two basic commands for plotting surfaces in 3-space: mesh and surf. The former produces a transparent “mesh” surface; the latter produces an opaque shaded one. There are two different ways of using each command, one for plotting surfaces in which the z coordinate is given as a function of x and y, and one for parametric surfaces in which x, y, and z are all given as functions of two other parameters. Let us illustrate the former with mesh and the latter with surf. To plot z = f (x, y), one begins witha meshgrid command as in the case of contour. For example, the “saddle surface” z = x 2 − y 2 can be plotted with >> [X,Y] = meshgrid(-2:.1:2, -2:.1:2); >> Z = X.ˆ2 - Y.ˆ2; >> mesh(X, Y, Z) The result is shown in Figure 5-6, although it looks much better on the screen since MATLAB shades the surface with a color scheme depending on the z coordinate. We could have gotten an opaque surface instead by replacing mesh with surf. 74 Chapter 5: MATLAB Graphics -2 -1 0 1 2 -2 -1 0 1 2 -4 -3 -2 -1 0 1 2 3 4 Figure 5-6 With the Symbolic Math Toolbox, there is a shortcut command ezmesh, and you can obtain a result very similar to Figure 5-6 with >> ezmesh(’xˆ2 - yˆ2’, [-2, 2], [-2, 2]) If one wants to plot a surface that cannot be represented by an equation of the form z = f (x, y), for example the sphere x 2 + y 2 + z 2 = 1, then it is bet- ter to parameterize the surface using a suitable coordinate system, in this case cylindrical or spherical coordinates. For example, we can take as param- eters the vertical coordinate z and the polar coordinate θ in the x-y plane. If r denotes the distance to the z axis, then the equation of the sphere becomes r 2 + z 2 = 1, or r = √ 1 − z 2 , and so x = √ 1 − z 2 cos θ , y = √ 1 − z 2 sin θ . Thus we can produce our plot with >> [theta, Z] = meshgrid((0:0.1:2)*pi, (-1:0.1:1)); >> X = sqrt(1 - Z.ˆ2).*cos(theta); Special Effects 75 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 Figure 5-7 >> Y = sqrt(1 - Z.ˆ2).*sin(theta); >> surf(X, Y, Z); axis square The result is shown in Figure 5-7. With the Symbolic Math Toolbox, parametric plotting of surfaces has been greatly simplified withthe commands ezsurf and ezmesh, and you can obtain a result very similar to Figure 5-7 with >> ezsurf(’sqrt(1-sˆ2)*cos(t)’, ’sqrt(1-sˆ2)*sin(t)’, . ’s’, [-1, 1, 0, 2*pi]); axis equal Special Effects So far we have only discussed graphics commands that produce or modify a single static figure window. But MATLAB is also capable of combining several 76 Chapter 5: MATLAB Graphics figures in one window, or of producing animated graphics that change with time. Combining Figures in One Window The command subplot divides the figure window into an array of smaller figures. The first two arguments give the dimensions of the array of sub- plots, and the last argument gives the number of the subplot (counting left to right across the first row, then left to right across the next row, and so on) in which to put the next figure. The following example, whose output appears as Figure 5-8, produces a 2 × 2 array of plots of the first four Bessel functions J n ,0≤ n ≤ 3: >> x = 0:0.05:40; >> for j = 1:4, subplot(2,2,j) plot(x, besselj(j*ones(size(x)), x)) end 0 10 20 30 40 -0.5 0 0.5 1 0 10 20 30 40 -0.4 -0.2 0 0.2 0.4 0.6 0 10 20 30 40 -0.4 -0.2 0 0.2 0.4 0.6 0 10 20 30 40 -0.4 -0.2 0 0.2 0.4 0.6 Figure 5-8 [...]... basic principles concerning the way MATLAB stores and displays graphics For most purposes, the discussion here will be sufficient But if you need more information, you might eventually want to consult one of the books devoted exclusively to MATLAB graphics, such as Using MATLAB Graphics, which comes free (in PDF format) with the software and can be accessed in the MATLAB Manuals” subsection of the “Printable... replay the movie when it is done 78 Chapter 5: MATLAB Graphics Customizing and Manipulating Graphics This is a more advanced topic; if you wish you can skip it on a first reading So far in this chapter, we have discussed the most commonly used MATLAB routines for generating plots But often, to get the results one wants, one needs to customize or manipulate the graphics these commands produce Knowing how... item 80 Chapter 5: MATLAB Graphics Text Properties in the Tools menu (in MATLAB 5.3), or else the menu item Figure Properties in the Edit menu (in MATLAB 6), can be used to change the font style and font size Change of Viewpoint Another common and important way to vary a graphic is to change the viewpoint in 3-space This can be done with the command view, and also (at least in MATLAB 5.3 and higher)... often more convenient, especially when one wishes to “experiment” with various changes, while editing a figure with MATLAB code is often required when writing M-files So the true MATLAB expert uses both techniques The figure window menus are a bit different in MATLAB 6 than in MATLAB 5.3 In MATLAB 6, you can zoom in and out and rotate the figure using the Tools menu, you can insert labels and text with the... Y4, ’b*’) Customizing and Manipulating Graphics 81 Here we would probably want the tick marks on the x axis located at multiples of π This can be done with the set command applied to the properties of the axes (and/or by selecting Edit : Axes Properties in MATLAB 6, or Tools : Axes Properties in MATLAB 5.3) The command set is used to change various properties of graphics To apply it to “Axes”, it has... addition, the low-level graphics commands line, rectangle, fill, surface, and image can be used to create new graphics elements within a figure window As an example of these techniques, the following code creates a chessboard on a white background, as shown in Figure 5-11: >> white = [1 1 1]; gray = 0.7*white; >> a = [0 1 1 0]; b = [0 0 1 1]; c = [1 1 1 1]; Figure 5-11 84 Chapter 5: MATLAB Graphics >> figure;... Manuals” subsection of the “Printable Documentation” section in the Help Browser (or under “Full Documentation Set” from the helpdesk in MATLAB 5.3 and earlier versions), or Graphics and GUIs with MATLAB, 2nd ed., by P Marchand, CRC Press, Boca Raton, FL, 1999 In a typical MATLAB session, one may have many figure windows open at once However, only one of these can be “active” at any one time One can find... Figure 5-10 -pi 0 pi 2pi 82 Chapter 5: MATLAB Graphics >> set(gca, ’FontName’, ’Symbol’) >> set(gca, ’XTickLabel’, ’-2p|-p|0|p|2p’) since in the Symbol font, π occupies the slot held by p in text fonts Full-Fledged Customization What about changes to other aspects of a plot? The useful commands get and set can be used to obtain a complete list of the properties of a graphics window, and then to modify... of a lot of numerical data with each frame of the movie Finally, make sure that while MATLAB executes the loop that generates the frames, you do not cover the active figure window with another window (such as the Command Window) If you do, the contents of the other window will be stored in the frames of the movie  MATLAB 6 has a new command movieview that you can use in place of movie to view the animation... out on the ray z = 0.5t, x = −0.5272t, y = −0.3044t, t > 0 ➱ In MATLAB, any two-dimensional plot can be “viewed in 3D,” and any three-dimensional plot can be projected into the plane Thus Figure 5-5 above (the helix), if followed by the command view(2), produces a circle Change of Plot Style Another important way to change the style of graphics is to modify the color or line style in a plot or to change . exclusively to MATLAB graphics, suchas Using MATLABGraphics, which comes free (in PDF format) with the software and can be accessed in the MATLAB Manuals”. and customizing them. You can get a list of MATLAB graphics commands by typing help graphics (for general graphics commands), help graph2d (for two-dimensional