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. The sides of your rounded edge squares are impressively flat. 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. Whit http://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 â(one ninth constant), which I claim should be named Halphen's.
It's the largest q for which the curve is convex.))
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.