TL;DR

Where I wanted to generate different PNG files with Gnuplot, each corresponding to a different mix of two functions, parametrically.

For a future post (I hope), I wanted to generate different images for showing the transition from the linear space to the transformed space in Fortune’s algorithm (see also the other post). And I wanted to use Gnuplot.

My first step has been to make the Gnuplot script parametric, which is easy:

 1 #!/usr/bin/env gnuplot
 2
 3 # Mathematics
 4 Ax = 0
 5 Ay = 0
 6 m = 0.9
 7 q = -1.0
 8 line(x) = m * x + q
 9 dist(x) = sqrt((Ax - x)**2 + (Ay - line(x))**2)
10 hyperb(x) = line(x) + dist(x)
11 r = (0.0 + n) / N
12 mix(x) = (1 - r) * line(x) + r * hyperb(x)
13
14 # Graphics
15 reset
16 set terminal pngcairo size 410,250 enhanced font 'Verdana,9'
17 set output filename
18 set style line 1 lc rgb 'red'  lw 2
19 set style line 2 lc rgb 'blue' pt 7
20 set key off
21 set yrange [-2:1.2]
22
23 ## Draw point
24 set label at Ax, Ay "A" point ls 2
25
26 ## Draw mix
27 plot [-1:1] mix(x) t 'mix' ls 1

Local version (without line numbers!).

As indicated by the comments, the first section (lines 4 through 12) include the mathematics for the curves and their mix:

  • lines 4 through 7 include the parameters for the A point (via Ax and Ay) and the line (via m and q);
  • line 8 defines the line function, using m and q;
  • line 9 is a helper function to calculate the distance of a point on the line from the reference point A;
  • line 10 calculates the hyperbole according to Fortune’s algorithm;
  • line 11 calculates r, that allows mixing line and hyperb together. It is supposed to range between 0 (all line) and 1 (all hyperb);
  • line 12 calculates the mix of line and hyperb, according to the ratio r.

The mix is just a linear combination of the two functions, according to a ratio r. This is calculated using the two parameters n and N, which are supposed to come from the outside; this allows passing the ratio as a numerator/denominator pair, which allows using integers from the outside.

The rest is graphics stuff. One important thing is that line 17 sets the name of the output file according to the filename variable, which is supposed to be set from the outside.

Invoking the script, then, requires setting n, N, and filename, like this:

# set at 10/25th of the mix, save in test.png
$ gnuplot \
   -e 'n=10' \
   -e 'N=25' \
   -e 'filename=test.png' \
   20200303-parametric.gnuplot

And I guess it’s all for today!