Parametric plot from a file II.

As I promised yesterday, we will take a closer look at the pie chart, once more, and see how we can utilise what we have learnt recently. I should point out here, that this is not the only way of plotting a pie from a file. If you feel like building your gnuplot from source, you can check out either the CVS tree, or the patch tracker, where you can find a patch that makes it possible to plot slices. You can see a demo here. But we will try a different route here.

First, here is our data file (it could be anything, really)

1 1 Dolphins
2 1 Whales
2 0 Sharks
3 0 Penguins
4 1 Kiwis
5 0 Tux

and here is our script

reset
unset key; set border 0; unset tics; unset colorbox; set size 0.6,1
set urange [0:1]
set vrange [0:2*pi]
set macro
sum = 0.0
ssum = 0.0
n = 0
PLOT = "splot 0, 0, 1/0 with pm3d"

count(x) = (ssum = ssum + $1, 1)
g(x,y,n) = \
sprintf(", \
u*cos((%.2f+%.2f*v)/ssum), \
u*sin((%.2f+%.2f*v)/ssum), \
%d @PL", x, y, x, y, n)

f(x) = (PLOT = PLOT.g(2*pi*sum, x, n), sum = sum+x, n = n + 1, x)

plot 'new_pie.dat' u 1:(count($1)), '' u 1:(f($1))

PL = "with pm3d"
set parametric; set pm3d map; 
eval(PLOT)

There is really nothing that we haven't discussed before: we set a couple of things at the beginning, but most importantly, the macro, and sum, ssum, and n. Then we define a string, PLOT, and two functions. One is to sum the values in our file (we need this, so that we can scale the full range of angles to two pi), and another one, that writes our PLOT command for later use. Note that the first plot in PLOT is actually empty, we plot 1/0. This seems a bit silly, doesn't it? Well, it does, but there is a good reason: successive plots must be separated by a comma, and if we have an empty plot at the very beginning, then we can put the commas before the plots, not after, and in this way, we needn't keep track of which plot we are actually processing. Remember, yesterday we used a separate counter, and an if statement, to determine, whether we need the comma, or not. This we can avoid here.
Next we call the two dummy plots, and finally, we evaluate our PLOT string. Oh, no! At the very end, we marvel in awe at the figure that we produced.

So far, so good, but what if we wanted to add labels, e.g., the value of the slice? That is really easy. All we have to do is to define a function that produces the label. Here is our updated script

reset
unset key; set border 0; unset tics; unset colorbox; set size 0.6,1
set urange [0:1]
set vrange [0:2*pi]
set macro
sum = 0.0
ssum = 0.0
n = 0
PLOT = "splot 0, 0, 1/0 with pm3d"
LABEL = ""

count(x) = (ssum = ssum + $1, 1)
g(x,y,n) = \
sprintf(", \
u*cos((%.2f+%.2f*v)/ssum), \
u*sin((%.2f+%.2f*v)/ssum), \
%d @PL", x, y, x, y, n)

lab(alpha, x) = sprintf("set label \"%s\" at %.2f, %.2f; ", \
x, 1.2*cos(alpha), 1.2*sin(alpha))

f(x) = (PLOT = PLOT.g(2*pi*sum, x, n), \
LABEL = LABEL.lab(2*pi*sum/ssum+pi*x/ssum, sprintf("%2.f", x)), \
sum = sum+x, n = n + 1, x)

plot 'new_pie.dat' u 1:(count($1)), '' u 1:(f($1))

PL = "with pm3d"
set parametric; set pm3d map; set border 0; unset tics; unset colorbox;
set size 0.6,1
eval(LABEL)
eval(PLOT)

We have an addition string, LABEL, which we initialise with the value "". Then we define a function that prints "set label ..." with the proper positions, and finally, we insert this function in the definition of f(x). Of course, once we called f(x) in the plot, we have to evaluate the string LABEL. So, this is what we get

This script can trivially be modified to print strings that are stored in our file. Watch just the following two lines

...
lab(alpha, x) = sprintf("set label \"%s\" at %.2f, %.2f centre; ", \
x, 1.2*cos(alpha), 1.2*sin(alpha))

f(x) = (PLOT = PLOT.g(2*pi*sum, x, n), \
LABEL = LABEL.lab(2*pi*sum/ssum+pi*x/ssum, stringcolumn($3)), \
sum = sum+x, n = n + 1, x)
...

and we are done. Here is the new pie

Now, you might wonder why we had three columns in our data file, if we didn't want to use it. Well, perhaps, we wanted, just haven't got time till now. So, what could we do with those ones and zeros in the second column? We will make the pie explode! It is really simple, we have to modify two lines in our last script

...
g(x,y,n,dx,dy) = \
sprintf(", \
%.2f+u*cos((%.2f+%.2f*v)/ssum), \
%.2f+u*sin((%.2f+%.2f*v)/ssum), \
%d @PL", 0.2*dx, x, y, 0.2*dy, x, y, n)

lab(alpha, x, r) = sprintf("set label \"%s\" at %.2f, %.2f centre; ", \
x, (1.25+0.2*r)*cos(alpha), (1.25+0.2*r)*sin(alpha))

f(x) = (PLOT = PLOT.g(2*pi*sum, x, n, \
$2*cos(2*pi*sum/ssum+pi*x/ssum), \
$2*sin(2*pi*sum/ssum+pi*x/ssum)), \
LABEL = LABEL.lab(2*pi*sum/ssum+pi*x/ssum, stringcolumn($3), $2), \
sum = sum+x, n = n + 1, x)
...

And here is the pie, when exploded

I should mention here that if you are not happy with the colours, it is really easy to help it: all we have to do is to modify the colour palette, using whatever colour combinations. We have covered a lot of material today. Till next time,
Gnuplotter

Plotting in 6 dimensions - parametric plot from a file

Today I would like to touch on a vast subject, so prepare for a long post. However, I hope that the post will be worthwhile, for I want to discuss something that cannot be done in any other way. In due course, we will see how we can use gnuplot to create parametric plots from a file. What I mean by that is the following: if you want to plot, say, 10 similar objects, whose size is determined by the first column in a file. Of course, there are cases, when one can manipulate the size, e.g., if there is a pre-defined symbol, we can use one of the columns in a file to determine the size of the symbol. As an example, we can do this

plot 'foo' u 1:2:3 with point pt 6 ps var

which will draw circles whose radius is given by the third column in 'foo'. This is all well, but it works for a limited number of cases only, namely, when there is a symbol to start out with. But what happens, if we want to draw an object that is not a symbol, e.g., arcs of a circle, whose angle is given by one of the columns in a file, or cylinders, whose height is a variable, read from a file. As you can guess from these two suggestions, what we will do is to draw a pie, and a bar chart. I understand that we have done this a couple of times before, but this time, we will stay entirely in the realm of gnuplot, and the scripts are really short. We just have to figure out what to write in the scripts. But beyond this, I will also show how we can plot in 6 dimensions. We will plot ellipses on a plane (first 2 columns), whose two axes are given by the 3rd and 4th axis, the orientation by the 5th, and the colour by the 6th. If you are really pressed for it, you can add three more dimensions: if you draw ellipsoids in 3D, take all three axes from a file, and also the orientation, that would make 9 dimensions altogether. Quite a lot!

So, let us get down to business! The first thing that I would like to discuss is the evaluate command. This is a really nifty way of shortening repetitive commands. Let us suppose that we want to place 10 arrows on our graph, and only the first coordinate of the arrows changes, otherwise everything is the same. Setting one arrow would read as follows

set arrow from 0, 0 to 1, 1

Of course, there are quite a few settings that we could specify, but this was supposed to be a minimal example. Then, the next arrow should be

set arrow from 1, 0 to 2, 1

and so on. What if we do not want to write this line a thousand times, and we do not want to search for the coordinate that we are to change, the first one, in this case? We could try the following

a(x) = sprintf("set arrow from %d, 0 to %d, 1", x, x+1)

This function takes 'x', and returns a string with all the settings and coordinates. So, we are almost done. The only thing we should do is to make gnuplot understand that what we want it to treat a(x) as a command, not as a string. Enter the eval command: it takes whatever string is presented to it, and turns it into a command. Thus, the following script creates 5 arrows, all parallel to each other, and consecutively shifted to the rigth

a(x) = sprintf("set arrow from %d, 0 to %d, 1", x, x+1)
eval a(0)
eval a(1)
eval a(2)
eval a(3)
eval a(4)

I believe, this is a much simpler and cleaner procedure, than this

set arrow from 0, 0 to 1, 1
set arrow from 1, 0 to 2, 1
set arrow from 2, 0 to 3, 1
set arrow from 3, 0 to 4, 1
set arrow from 4, 0 to 5, 1

I should mention here that if chunks of a command are the same, another method of abbreviating them is to use macros. Those are disabled by default, so first we have to set it. Then it works as follows

set macro
ST = "using 1:2 with lines lt 3 lw 3"
plot 'foo' @ST, 'bar' @ST 

i.e., the term @ST is expanded using the definition above, therefore, this plot is equivalent to this one

plot 'foo' using 1:2 with lines lt 3 lw 3, 'bar' using 1:2 with lines lt 3 lw 3

but the previous one is much more readable. I would also say that using capitals for the macros is probably not a bad idea, because then they cannot be mistaken for standard gnuplot commands. This much in the way of macros!

So, we have the evaluate command, and we have a new concept for functions. Then let us take a closer look at the following code

a(x) = sprintf("set arrow from %d, 0 to %d, 1;\n", x, x+1)
ARROW = ""
f(x) = (ARROW = ARROW.a(x), x)
plot 'foo' using 1:(f($1))

and let us suppose that our file 'foo' contains the following 5 lines

1
3
5
7
9

After plotting 'foo', the string 'ARROW' will be the following

set arrow from 1, 0 to 2, 1;
set arrow from 3, 0 to 4, 1;
set arrow from 5, 0 to 6, 1;
set arrow from 7, 0 to 8, 1;
set arrow from 9, 0 to 10, 1;

I.e., we have a string, which contains instructions for setting 5 arrow. If, at this point, we simply evaluate this string, all 5 arrows will be set. Therefore, we have found a way of using a file to set the coordinates of an arrow. (N.B., if it was for the arrows only, we wouldn't have had to do anything, since there is a plotting style, 'with vector', as we discussed some weeks ago.)

We will use this trick to create a parametric plot, taking parameter values from a file, first plotting the ellipses! Again, we have got to create some dummy data, and since we now need 6 columns, we will use the errorbars

reset
f(x) = rand(0)
set sample 50
set table 'ellipse.dat'
plot [0:10] '+' using (20*f($1)):(20*f($1)):(f($1)):(f($1)):(3.14*f($1)):(f($1)) w xyerror
unset table

which will produce 6 columns and 50 lines. Having produced some data, let us see what we can do with it. Here is our script:

PRINT(x, y, a, b, alpha, colour) = \
sprintf("%f+v*(%f*cos(u)*cos(%f)-%f*sin(u)*sin(%f)),
%f+v*(%f*cos(u)*sin(%f)+%f*sin(u)*cos(%f)),
%f with pm3d", x, a, alpha, b, alpha, y, b, alpha, b, alpha, colour)
PLOT = "splot "
num = -1
count(x) = (num = num+1, 1)
g(x) = (PLOT = PLOT.PRINT($1, $2, $3, $4, $5, $6), \
($0 < num ? PLOT=PLOT.sprintf(",\n") : 1/0))
plot 'ellipse.dat' u 1:(count($1))
plot 'ellipse.dat' using 1:(g($1))

unset key
set parametric
set urange [0:2*pi]
set vrange [0:1]
set pm3d map
set size 0.5, 1
eval(PLOT)

First, we have the definition of a print function that looks rather ugly, but is quite simple. We want to plot

a*v*cos(u), b*v*sin(u), colour

where a, and b are the axes of the ellipse, and colour is going to specify, well, its colour. However, we want to translate the ellipse to its proper position, and we also want to rotate it by an amount given by the 5th column, so we have to apply a two-dimensional rotation on the object. Therefore, we would end up with a function similar to this

x+v*(a*cos(u)*cos(alpha)-b*sin(u)*sin(alpha)), y + v*(a*cos(u)*sin(alpha)+b*sin(u)*cos(alpha)), colour

Now you know why that print function looked so complicated! After this, we define a string, PLOT, that we will expand as we read the file. But before that, we have to count the lines in the file. The reason for that is that successive plots must be separated by a comma, but there shouldn't be a comma after the last plot. So, we just have to know where to stop placing commas in our string. Then we define the function that does nothing useful, but concatenates the PLOT string as it reads the file. Here we use the number of lines that we determine in a dummy plot. At this point we are done with the functions, all we have to do is plotting.
First we count, then plot g(x). At this point, we have the string that we need. We only have to set up our plot. Remember, we have a parametric plot, where the range of one of the variables is in [0:2*pi], while the other one is in [0:1]. Easy. Then we just have to evaluate our plot string, and we are done. Look what we have made here: a six-dimensional plot!

I think that this script is much less complicated, than many that we have discussed in the past. Short and clear, thanks to the eval command, and the new concept of functions. Besides, we pulled off a trick that was impossible by other means. I started out saying that we will create bars and pie. I believe, having seen the trick, it should be quite simple now, but in case you insist on seeing it, I will discuss it in my next post.