----- Original Message ---- From: R. William Gosper <rwg@osots.com> To: math-fun@mailman.xmission.com Sent: Monday, February 19, 2007 10:31:59 PM Subject: Re: [math-fun] Simplest Ovals (WAS: sections of quadratic surfaces)

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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". ------- If the periodic continuous function f(t) has continuous r-th derivative, then the Fourier series coefficients are O(1/n^(r+2)). Given a closed curve C in the complex plane, and nonnegative integer r, there is a function f(t) whose graph is C and which has continuous r-th derivative. ------- Are there smooth speed functions with no counterrotors (negative harmonics)? --------- A simple closed curve is the boundary of a simply connected region R. By the Riemann mapping theorem, there is a conformal map f from the interior of the unit disk onto R, and f(z) = sum(a[n] z^n, n>=0) since the map is analytic in the unit disk. The curve C is traced when z = exp(it), and so there are no counterrotors. A nonsimple closed curve can often be obtained by deforming a simple curve onto sheets of a Riemann surface. A conformal mapping from the unit disk should still exist, and so the conclusion holds for this case as well. I'm not sure how pathological the curve must be before this construction breaks down. Gene --------- --rwg PS, This problem ought to be "trivial" with kappa(s) (curvature(arclength)) notation. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun ____________________________________________________________________________________ Finding fabulous fares is fun. Let Yahoo! FareChase search your favorite travel sites to find flight and hotel bargains. http://farechase.yahoo.com/promo-generic-14795097