clothoid weekend update

For context, see past posts in the *curve-fitting* category that I just created. To recap:

The curves I’ve been drawing are the paths made by a point moving at constant speed at an angle which is a piecewise quadratic function of path length. Curvature, the first derivative of angle, is continuous.

Such a path that hits a given sequence of dots is fully determined if it loops, but otherwise it has two degrees of freedom. For any angle and curvature at the starting dot, there is a quadratic coefficient that lets the path reach the next node, and likewise for the next.

My current code starts with an estimate for the length of each segment (between two dots) and the angle at its midpoint, and uses these basis functions to fit those angles: a constant, a linear function, and a family of “solitons”: piecewise quadratics, zero outside a sequence of four dots, discontinuous in the second derivative at each of those dots. For n segments, there are n-2 solitons, so the constant and linear functions are needed to consume the last two degrees of freedom.

Eventually I noticed a flaw in this scheme: the curvature of the resulting path is the same at both ends, namely the slope of the linear component, because the solitons contribute nothing to it. That’s appropriate for āCā, but wrong for plenty of other strokes; in āSā the end curvatures ought to have opposite sign.

The next thing I’ll try is a least-squares quadratic fit to the whole sequence, then fit the residues with solitons as before. That should be an improvement but it’s not ideal; curvature is a local feature. Perhaps I’ll think of something better later.

un-meander

Here, each curve hits alternate dots: first exactly (above), then with offsets pushing it toward the other curve. Below is the result of eight iterations.

With enough iterations, the top of ‘s’ eventually gets a more symmetrical arch, as the change in curvature is spread more evenly.

But many runs get stuck in the fitting phase, and I don’t yet know why. Attacking one likely flaw didn’t help.

meander

(Previously: 2014, 2011, 2010; also, less closely related, 2015)

I tried to smoothen a stroke by shifting each dot toward the Euler spiral (aka clothoid, aka Cornu spiral) determined by its four nearest neighbors. That didn’t work so well: small wiggles were removed, but big ones were magnified.

( . . more . . )

mutant dragons

In 2007 I thought of a pretty way to paint a square so that all pixels are different, but similar colors are clustered. For each pixel, set x,y to its coordinates; if their sum is odd, set the low bit of one of the color channels to 1. Replace x,y with (-x+y)>>1, (-x-y)>>1; this has the effect of rotating the grid by 3/8 turn and shrinking it by a factor of √2, so that the former even points, which formed a larger oblique grid, now fall on the original grid, and the odd points have their new half-coordinates truncated away. Repeat until a bit has been assigned to each bit of the three color channels.

(More concisely: considering the pixel’s coordinates as a complex number, express it as a bit string in base (-1+i).)

Colors that match in their higher bits form twindragon fractals, thus:

In 2012, I thought: what if the rotation alternates clockwise and counterclockwise?

A bit on the boring side.

But in 2017, I thought: what about less trivial sequences? ( . . more . . )

how (some) fireflies do it

Fireflies in Borneo have a wonderfully simple and distributed way of synchronizing their blinks.

If I had the skill I’d make a screensaver of it. A couple of ways to play with the concept:

I wouldn’t expect all bugs to have *exactly* the same period, but how much variation is tolerable? What if there are two populations, indistinguishable except that their periods differ by an irrational factor?

What if each bug has a different hue, and responds only to others that are near on the color wheel (perhaps only in one direction)? Might a stable cycle result, rather than synchrony of all?

what, more links?

Hm, the first two links here have been lying around for five years; guess I ought to shove them out.

Pascal-Emmanuel Gobry on restructuring the banks

“Zomia”, a large region in Asia that was effectively stateless until recently

James Leroy Wilson on The Limits of Utilitarianism. The payoff is near the bottom.

points problems pointers problems

Two-fifths of the links in my sphere arrangements page were broken. Ouch! I found new addresses for a few of them, and used the Wayback Machine for the rest.