On Thu, Aug 28, 2014 at 12:45 AM, Bill Gosper <billgosper@gmail.com> wrote:

Henry>I wanted an approximation that was entire over the complex plane, so the straight lines and discontinuities would have to be fudged.

If I take the image, convert it to black-and-white, and erode the boundaries to a single pixel width, I will get a "continuous" curve that goes around the origin twice.

If I consider this curve in (r,theta) coordinates, I can now consider the Fourier transform. I'd like to reset the origin in such a way that the "fundamental" is zero -- i.e., the lowest frequency (which takes 2 revolutions) has a coefficient of zero. (There are other possible locations for the origin: the balancing point, the center of gyration, etc.)

That having been done, I could alternatively use (log(r),theta) coordinates, which have some nicer properties, because it is the actual complex log function.

Fourier analysis can approximate the discontinuous changes in curvature; hopefully modifying these coefficients a little will remove these discontinuities with a minimum of ringing.

Ringing only happens at true discontinuities (e.g.http://www.tweedledum.com/rwg/gibbs.htm )

A nice example of going around twice, mixing circular arcs and straight segments, is to draw, with a single set of amplitudes and phases (for which I have closed forms) alternately an equilateral triangle and its inscribed circle:gosper.org/Incirc_1.mp4 (The file isn't corrupt--download it.)

The sides of your rounded edge squares are impressively flat. <HGB

At 03:48 PM 7/30/2014, Bill Gosper wrote:

HGB> My approach would be to take the Fourier transform of the polar plot,

rwg>Defined as piecewise linear|circular?

and then simplify to a small number of coefficients.

Of course, it will take 2 circuits around the origin to complete the figure.

The sides are perhaps a little too flat, which will blow up the number of coefficients required.

I guess if you transform [0,4pi) to [0,1) you could use some sort of Cheby polynomial approximation.

It would be nice to be able to transform it back to p(x,y)=0, where p is a polynomial in x,y.

At 02:56 PM 7/29/2014, Whitfield Diffie wrote: HGB>Anyone game for an analytic function to approximate Airbnb's new logo? They should manufacture a paper clip and give it out at trade shows. Whithttp://www.underconsideration.com/brandnew/archives/airbnb_logo_detail.png

rwg> You can get some impressively straight segments with theta fcns: gosper.org/halph.pdf (10MB?!) or see http://gosper.org/thetpak.html (scrolled 69%). (.3281... is

√(one ninth constant), which I claim should be named Halphen's.

It's the largest q for which the curve is convex.))

As I mentioned here before, as 0<q<1, the curve continuously deforms from a tiny "circle" to a big "+" sign, sweeping out a fairly convincing spaceplane: http://gosper.org/sst.png --rwg

Two more mystery constants associated with this spaceplane: Its volume (4.231765644191597 ± ? (Mma 10.0.2 crashed when I asked for more digits), And its "fattest" (cross-sectional area, vs squarest) q, Out[87]= {q -> 0.43939962899109219485183075448731} which ISC guessed was In[81]:= 1/Log[3]^(7/4)/2^(1/12)/E^(3/5) (nice try). Pix: gosper.org/halphen.pdf The first graph (with strangely varying curvature) is cross sectional area vs q. I had to downgrade to 10.0.1 to make this pdf. This is the first time I've ever noticed an effect of a minor version increment, and it was to render all text huge and illegible! --rwg

But a problem with this logo is that it switches curvatures discontinuously. <rwg

Another possibility: modulate the arcspeed (slow down for curves, speed up for straightaways) before taking the Fourier.