Visualizations made easy with gnuplot

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This article, written by Lee Philips the author of gnuplot Cookbook contains the following :

 

  • Making a surface plot
  • Using coordinate mappings
  • Coloring the surface
  • Making a contour plot
  • Making a vector plot
  • Making an image plot or heat map
  • Combining contours and images
  • Combining surfaces with images
  • Plotting a path in 3D
  • Drawing parametric surfaces

Making a surface plot

A surface plot represents the dependent quantity z, which depends on the two independent variables x and y, as a surface whose height indicates the value of z.

 

The previous figure is a perspective drawing of a surface representing the Bessel function J0(r), where r is the distance from (x=0, y=0). The height of the surface shows the value of J0, given on the vertical axis (unlabeled in this figure, but usually called z). The other two (unlabeled) axes defning the plane above which the surface is drawn are the x and y axes.

 

 

How to do it…

The following code listing is the script that coaxed gnuplot into making the previous figure:

 

set isosamples 40
unset key
set title “J_0(r^2)”
set xrange [-4:4]set yrange [-4:4]set ztics 1
splot besj0(x**2+y**2)
set view 29,53 #Done implicitly by mousing.
set term pngcairo mono enhanced
set out ‘bessel.png’
replot

 

How it works…

There are several new commands in this recipe. The set isosamples command sets the isoline density. This is analogous to the set samples command when making 2D plots, but it sets the number of lines used in forming the surface. The number of isosamples can be set independently in each direction; if one number is specifed, it is used for both directions. The default of 10 usually creates surfaces that are far too coarse to be useful.

 

Turning to the second of the highlighted commands, the splot command is the 3D version of our old friend the plot command (it probably initially stood for “surface plot”, but now can do several things besides plot surfaces, as we shall see in the rest of this article). It expects a function of x and y rather than x alone. Although we are interested in plotting something that has the type of symmetry that would be most conveniently expressed in polar (spherical or cylindrical) coordinates, these geometries are not available for function plots in 3D in gnuplot. (They are available through the set mapping command for data plots, as we shall see later in this article.) Therefore in such cases, we are required to convert our expressions to the rectangular coordinate system. Instead of what we would call r in a cylindrical coordinate system, here we use the equivalent x**2 + y**2.

 

In this recipe, we would like to illustrate, as far as possible, the interactive approach to creating a fnal 3D plot. The next highlighted line, beginning with set view, can be entered on the command line or included in a script. The view is the orientation in degrees of the perspective drawing of the 3D plot. Naturally, it does arise in 2D. It is diffcult to determine what is the most useful view for a particular plot without looking at it and experimenting with it; therefore, even if our fnal product is intended to be a fle, a common workfow is to frst create the plot using an interactive terminal (x11 or wxt). Then we rotate the plot with the mouse, and possibly scale and zoom it using the middle mouse button, until we arrive at the desired appearance. This is what we mean by the comment in the set view command. Now we can reset the terminal to the fnal output device that we need, specify the output fle, and simply say replot. The view and scaling at which we left the interactive plot is retained as a set of global settings and will be refected in our fnal output fle. These settings are also displayed at the bottom of the interactive plot window, so we can record them if we are going to make similar plots in the future, or want a set of plots to be drawn with the same settings. Note that we also redefned the ztics value. This is because when the plot is tilted to the fnal view angle that we chose, the perspective causes the tic labels on the vertical axis to be crowded together; this is a common problem with 3D plots, and taking manual control of the tics on the z-axis is the solution.

 

There’s more…

Following is the same plot with one setting changed (aside from a slight adjustment in the view angle):

 

While the frst plot was essentially a wireframe that we could see through, this version has the appearance of a solid, opaque surface. All we need to do is to say set hidden3d. This, which only works when lines or linespoints are being used, makes the surface appear opaque by removing from the plot any part of the surface, other surfaces, and other plot elements such as the axes and tic labels, that are behind the surface from our point of view. The underside of the surface is shown in a contrasting color with a color output device, but the two sides of the surface are not distinguished in monochrome. The name of the setting refers to the technique of hidden line removal; gnuplot is justly famed for the quality of its hidden line removal algorithm, and is one reason this program is so well regarded for its 3D plotting ability.

 

Using coordinate mappings

It is possible, when making 3D plots from data fles, for the data to be interpreted in spherical or cylindrical coordinates rather than the default Cartesian system. For details, type help set mapping. We will give an example of using the cylindrical mapping to conveniently draw a shape with cylindrical symmetry.

 

The previous figure is a perspective view of a surface that somewhat resembles a Christmas tree ornament. The relevant feature of this surface is that it has rotational symmetry around the z (vertical) axis, which means that it is most naturally expressed in cylindrical coordinates.

 

How to do it…

Try the following script:

 

set mapping cylindrical
unset tics
unset border
set hidden
set xrange [-pi : pi]set yrange [-pi : pi]set zrange [0 : pi]set iso 60
unset key
splot ‘++’ using 1:2:(sin($2)) with lines

 

How it works…

There are several new ideas used in this recipe. Breaking it down, these are:

 

The set mapping command

The frst, highlighted line contains the new command that is the subject of this recipe. When the default Cartesian (x-y-z) coordinate system is changed to cylindrical then the columns of data read in during a data plot are interpreted as θ-z-r, where θ is the angular coordinate, z is the vertical coordinate, and r is the radius. A spherical mapping is also available and explained in the gnuplot online help (help set mapping). If the data fle only has two columns, then the plot is drawn with r = 1.

 

In our example we don’t want to plot from a data fle, however. We want to plot a function given directly in the script. This presents us with a problem, as gnuplot does not support cylindrical or spherical plots of functions in 3D. The solution is to use one of gnuplot’s pseudofles.

 

The ++ pseudofle

The “++” pseudofle creates rows of imaginary data with three columns x-y-z unless we change the coordinate mapping, which of course in this example we have. Setting the mapping to cylindrical means that the fctitious data columns will be interpreted as θ-z-r.

 

Now to plot a function, we use the using notation applied to the imaginary columns of data. We’ve done this in the fnal line of the script, where we plot the sine of the second column (z).

 

To clarify the use of “++” when plotting surfaces, note that, in Cartesian coordinates, the two commands “splot sin(x)+cos(y)” and “splot ‘++’ using 1:2:(sin($1)+cos($2)) with lines” produce exactly the same plot.

 

Coordinate ranges

We have also established ranges for all variables in the set xrange and two other commands following it. The ranges for the polar coordinates are taken from the corresponding Cartesian coordinates, that is, when we set the xrange, we are setting both the range of the x-axis displayed on the plot and the range of the variable θ in the cylindrical coordinate system. It is mandatory to set xrange and yrange when using the “++” flename.

 

This mixing of the coordinate system in which the function is calculated and the Cartesian system in which it is displayed can be confusing, but the example shows a strategy, which should make it possible to get predictable results. Setting the xrange and yrange as we’ve done puts the r = 0 point in the middle of the graph and prevents part of the plot from being cut off. It also sets up a full rotation of the angular coordinate over a range of 2 p.

 

If we wanted to plot, say, our shape with half of it sliced off by a vertical plane, the easiest way to do this is not to fddle with the coordinate ranges, but to apply a transformation to one of the fctitious data columns: splot ‘++’ using ($1/2):2:(sin($2)) with lines, will do the trick without any surprising side effects. In this example the underlying angular coordinate (column 1) still passes through a full rotation, but we’ve divided it in half without changing the way the figure is projected onto the Cartesian display. Note that the 60 isolines will still be used in our reduced angular range, so we might want to set iso to a smaller value.

 

Completing the picture

We’ve eliminated all of the graph adornments (unset tics, unset border, unset key) so we will be left with only the surface. The isosamples are set to give a suffciently smooth surface drawing that is nevertheless not too crowded with isosurface lines (see the previous recipe). set hidden ensures that we shall see only the outer surface of the shape.

 

Coloring the surface

The wireframe splot with hidden line removal that we covered in the frst recipe of this article, Making a surface plot, gives the visual impression of a solid surface. The numerical value encoded into the surface’s height can be visually estimated, roughly, by the perspective provided by the isolines in conjunction with the tics on the vertical axis. But gnuplot also has a way to draw real solid surfaces whose height is indicated by color or shade.

 

The previous figure shows the same mathematical function plotted in the frst recipe in this article (Making a surface plot). Now the numerical value of the function at any point is indicated by both the height of the surface and its shade; the surface is now drawn as an opaque membrane rather than as a network of curves.

 

How to do it…

 

To produce the previous figure, run the following in gnuplot:

 

set isosamples 100
set samples 100
unset key
set title “J_0(r^2)”
set xrange [-4:4]set yrange [-4:4]set ztics 1
unset surface
set pm3d
splot besj0(x**2+y**2)

The surface will be drawn with a palette of colors when a color output device is being used and with a grayscale palette when using a monochrome terminal.

 

 

How it works…

If you compare the previous script with the one in the Making a surface plot recipe at the beginning of this article, you will see that the only signifcant difference is the highlighted line. The pm3d mode colors the imaginary surface being plotted according to its height or z-value at every point, with the mapping between the height and the color or shade determined by the palette, which we shall discuss in some more detail shortly.

 

The other modifcations are to increase the number of isolines, in order to get a smoother surface, and to turn off the drawing of the individual isolines themselves with the command unset surface. We also need to set the sample frequency; generally we want this to be equal to the isosample frequency. In pm3d mode, the two orthogonal sets of isolines are drawn with two different spacings given by the two parameters. Although the gnuplot manual claims that the global hidden3d setting does not affect pm3d surface plots, it in fact seems to, and should not be turned on, as it appears to slightly degrade the drawing quality.

 

There’s more…

Sometimes we want both a colored surface and a set of isolines; in fact, this can often be the clearest type of quantitative 3D plot. The way to achieve the highest quality in this type of graph is to use the hidden3d linestyle option to pm3d, as we do in the following script:

 

set iso 30
set samp 30
unset key
set title “J_0(r^2)”
set xrange [-4:4]set yrange [-4:4]set ztics 1
unset surf
set style line 1 lt 4 lw .5
set pm3d at s hidden3d 1
splot besj0(x**2+y**2)

This requires us to defne a user linestyle. Then the linestyle is referred to in an option to the set pm3d command. This will cause the isolines to be drawn using lines in this style, which allows us to have them in any color, thickness, or pattern supported by our terminal. Further, the isolines will be drawn with hidden line removal, so they will appear to be embedded in the opaque surface. As before, the global hidden3d option should not be turned on.

 

Note that we’ve also reduced the sample and isoline frequency, to keep our plot from being too crowded with isolines. (The at s component of the set pm3d command means at surface.)

 

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