Dragon Curve Formula

Question

Can someone please explain to me this formula?

Answer 1

One of the simplest descriptions of the dragon curve is that it's the union of two copies of itself, suitably transformed (scaled, rotated, and translated). The two functions you're given then represent the transformations that carry the 'full-sized' dragon curve onto the two subcurves that make it up.

So what good do they do? Well, both of the transformations listed are contractions; their determinant is less than one, which means that they 'shrink' any region they're applied to. This means, in particular, that you can start with any point — not just one on the curve — apply enough of these transformations to it, and wind up with some point on (or at least arbitrarily near) the curve. And if you do this process often enough, say by following this procedure:

choose (x,y) randomly between, say, -10 and 10
for (maybe 20 iterations) {
    randomly choose f=f1 or f2
    set (x,y) = f(x,y) using the chosen f()
}
plot the final (x,y)

...then after several thousand iterations of this you'll have a pretty good visual approximation of the curve. For more on how this works, the magical phrase is Iterated Function System.