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 √(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.