Re: [math-fun] Simplest Ovals (WAS: sections of quadratic surfaces)
An interesting family of ovoids (a word I prefer to oval, cuz oval has a deeply ingrained casual meaning) is given by applying the transformation (r,theta) -> (r^c, theta) (c > 1) to the circle r= cos(theta), i.e., with radius = 1/2 and center at (1/2,0). The resulting curve has the equation r = cos(theta)^c When c = p/q for integers p,q > 0 with q odd, then the curve is given by the polynomial equation (x^2+y^2)^((q+1)/2) = x^p. E.g., p/q =3/1 gives a reasonable ovoid. Still, "Moss's oval" ( http://mathworld.wolfram.com/MosssEgg.html ) looks to me more like an egg than does the above curve for any values of p and q. --Dan
On 2/19/07, Daniel Asimov <dasimov@earthlink.net> wrote:
Still, "Moss's oval" ( http://mathworld.wolfram.com/MosssEgg.html ) looks to me more like an egg than does the above curve for any values of p and q.
Which is in fact exactly the construction earlier extolled by David Cantrell. WFL
On 2/19/07, Fred lunnon <fred.lunnon@gmail.com> wrote:
On 2/19/07, Daniel Asimov <dasimov@earthlink.net> wrote:
Still, "Moss's oval" ( http://mathworld.wolfram.com/MosssEgg.html ) looks to me more like an egg than does the above curve for any values of p and q.
Which is in fact exactly the construction earlier extolled by David Cantrell.
Indeed it is! Thanks for the reference, Dan. I had no idea that MathWorld had such an entry. Now I need to find a copy of Mathographics in hopes of getting information about Moss... David
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.
Cheap and smooth: r=(5+cos(t))^2+(3+cos(2*t))^2 . Suppose we made a piecewise circular-cylindrical funhouse mirror. Could it put corners on its (virtual) image of a smooth object? --rwg
On 2/21/07, R. William Gosper <rwg@osots.com> wrote:
Cheap and smooth: r=(5+cos(t))^2+(3+cos(2*t))^2 .
Looks more like a squash or pear to me --- must be some funny-shaped birds in your part of the world, Bill!
Suppose we made a piecewise circular-cylindrical funhouse mirror. Could it put corners on its (virtual) image of a smooth object? --rwg
The tangent plane would have to change discontinuously for this to happen. WFL
Cheap and smooth: r=(5+cos(t))^2+(3+cos(2*t))^2 .
WFL>Looks more like a squash or pear to me Whoa, nonconvex even? It looks quite Mossy when I plot it with Macsyma, equalscale:true, but I don't know how to get equalscale in Mma, which gives me various avocados. (Are there any birds with nonconvex eggs?)
--- must be some funny-shaped birds in your part of the world, Bill! Amen. I recently rounded a corner on which calmly browsed an East African crested crane, perhaps in town to see "The Last King of Scotland." Or maybe it just wanted some action with an American whoopee crane. --rwg
participants (4)
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Daniel Asimov -
David W. Cantrell -
Fred lunnon -
R. William Gosper