One application I thought of for this is object motion in games: I tried it out by writing this little Canvas game, where the comets follow the curve's trajectory. The differences in plot density along the curve create natural-looking comet tails.
A variation on the algorithm, noted by A.M. Andrew, sorts the set lexicographically and finds the upper and lower hull chains separately. This 'monotone chain' version is often preferred, since it's easier to do robustly.
But as you can see from the comment, the series converges very slowly!
Here's a simple method I learned from Gareth Rees for finding the intersection (or parallel / collinear status) of two line segments. Rees credits Ronald Goldman, though I'd imagine the technique goes further back.
Continuing the theme of strange attractors, here's the well-known one embedded in the Ikeda map. The thumbnail at left shows the central 'vortex'
of the attractor, and links to a larger viewport.
In these images, I've plotted the real and imaginary components along the x and y axes respectively. But the more popular way to visualize this attractor adds an extra parameter to the system and is expressed in trigonometric functions. Such adaptation of the code below yields these results.
French astronomer Michel Hénon reported on this strange, fractal attractor in 1976. Since then, it has been among the most studied examples of chaotic dynamical systems.
For a round of code golf, I wrote this spare implementation of Chris Langton's remarkably simple universal computer. If you want amenities like pause, random starting pattern or even quit, check out this more complete version.
Like Barry Martin's 'Hopalong' fractal, this dynamical system from John Connett was first published in Scientific American in 1986. This demo is interactive: successively clicking two points specifies a rectangle to zoom into. Doing so, you'll see that the system isn't actually a fractal. Instead of self-similarity, deep zooms reveal peacock-like images.
This pattern generator, discovered by Barry Martin, was nicknamed 'Hopalong' when Scientific American introduced it in their September '86 issue.Clicking the window adjusts the viewport position; there is also an alternate version with color and animation.
Also, for a certain Rubyist friend, I wrote another lazily-evaluated, colored and animated implementation in Ruby.
It took me some experimentation to figure out how to color this derivation of the logistic map; I'm still not quite sure how the hues should scale as you zoom. But the bi-tonal method shown below works well enough to produce the image at left - click it for more detail.
What programmer hasn't at some point written an implementation of Benoit Mandelbrot's great discovery, the most famous fractal in the world? Here's my own minimal version, with the simplest possible coloring scheme. To interact with it, just click any two points: the window will zoom in on the rectangle they define.
This 'lava lamp' is actually a simple cyclic cellular automaton; the CA rule is courtesy of Jason Rampe. In keeping with the spirit of Haskell, I chose to implement it with a hashmap of points rather than an array. Needless to say, that's not practical; but this is just a demonstration.
This little program generates random, undirected graphs without loops or isolated vertices (though the graph is not necessarily connected).
I have found it useful to generate random inputs for testing graph algorithms.
