Still, "Moss's oval" ( http://mathworld.wolfram.com/MosssEgg.html ) Ach, that's what I was calling van Zwolle's. Here's a truncated Fourier expansion which is very close, yet infinitely differentiable:
sqrt(2) cis(6 t) sqrt(2) cis(4 t) cis(3 t) (3 sqrt(2) - 1) %pi cis(t) - ---------------- + ---------------- - -------- + -------------------------- 15 6 3 2 sqrt(2) cis(- 2 t) sqrt(2) cis(- 4 t) + cis(- t) - ------------------ + ------------------ 3 10 Some of these terms can probably be Remezed out without visible damage. (I once sent this list a note on complex Remez.) Note the surprising presence of counterrotors, presumably due to my lazy choice of traversal speeds: constant dtheta/dt on all four arcs, meaning instant ac/deceleration at each arc boundary. More generally, these "four point eggs" are determined by the radii of the two endcaps (r1 and r2), and the angles they span (t1 and t2), which determine the radius and span of the arc joining them. The fully general Fourier series is then oo ==== '' \ - 2 (r2 - r1) ( > cis(n t) (sin((n - 1) t1) sin(t2) / ==== n = -oo n - (- 1) sin(t1) sin((n - 1) t2))/((n - 1) n))/(sin(t2) - sin(t1)) 2 (r2 - r1) (t1 cos(t1) sin(t2) - %pi cos(t1) sin(t2) - sin(t1) t2 cos(t2)) + --------------------------------------------------------------------------- sin(t2) - sin(t1) - 2 cis(t) (((r2 - r1) t1 - %pi r2) sin(t2) + (r2 - r1) sin(t1) t2 + %pi r1 sin(t1))/(sin(t2) - sin(t1)), where sum'' means skip n=0 and n=1. Note the (n-1)n in the term denominator. Gene Salamin once convinced me that there are alternate speed functions capable of putting arbitrarily high order polynomials in that denominator, hence arbitrarily rapid convergence of the Fourier series. But this does *not* mean you need arbitrarily few terms! The actuality is that the higher degree you seek, the longer the series diddles around before "flooring it". Are there smooth speed functions with no counterrotors (negative harmonics)? --rwg PS, This problem ought to be "trivial" with kappa(s) (curvature(arclength)) notation.