Fred supplied pictures of ovals of Greg Fee's and Bill Gosper's, which I've added to http://www.tiac.net/~sw/2007/02/Steve_Gray_oval --Steve
Fred supplied pictures of ovals of Greg Fee's and Bill Gosper's, which I've added to http://www.tiac.net/~sw/2007/02/Steve_Gray_oval Gack! You'll never pin that avocado on me! That's not equalscale. Try gosper.org/cheapegg.gif . wfl>Correct --- I hadn't meant to imply that the curvature actually went negative --- just that it looked as if it was thinking about it! [Also I cheated, and tried varying Bill's constant coefficients ...] So did I. Interesting problem: Suppose we declare the Moss/Zwolle egg to be Platonically ideal, and seek the "closest" approximation of the form (a+b*cos(t))^2+(c+d*cos(2*t))^2. How do we measure the closeness of two curves? It's not the same as the closeness of two functions, because the curves are parameterized differently. We don't want the answer to change if we simply change the speed with which we trace one of the curves. I'm tempted to resort to a double prodigal(qx,qy). --rwg
On 2/23/07, R. William Gosper <rwg@osots.com> wrote:
Fred supplied pictures of ovals of Greg Fee's and Bill Gosper's, which I've added to http://www.tiac.net/~sw/2007/02/Steve_Gray_oval Gack! You'll never pin that avocado on me! That's not equalscale. Try gosper.org/cheapegg.gif .
Oeuf! abject apologies --- Maple default caught me out again. I'll send Steve a correction PDQ. [But it still has flat spots!]
So did I. Interesting problem: Suppose we declare the Moss/Zwolle egg to be Platonically ideal, and seek the "closest" approximation of the form (a+b*cos(t))^2+(c+d*cos(2*t))^2. How do we measure the closeness of two curves? It's not the same as the closeness of two functions, because the curves are parameterized differently. We don't want the answer to change if we simply change the speed with which we trace one of the curves.
Maybe integrate the area of the region between them, made non-negative throughout, and minimised over all possible isometry (or similarity) transformations? There's quite a lot of work been done on this by computer vision people --- you need to do something similar in order to recognise which of several possible templates fits an observed object best. But a lot of it uses invariants --- I don't think they help here.
I'm tempted to resort to a double prodigal(qx,qy).
Come again? Fred Lunnon
rwg>> I'm tempted to resort to a double prodigal(qx,qy). wfl>Come again? prodigal:integral :: geometric mean:arithmetic mean. prodigal :=e^integral(log). I was trying to create a surface e^-f^2, 1 on the curve, dropping sharply to 0, where f is 0 on the curve by virtue of being some prodigal of distance from it. But this was stupid. prodigal(distance from curve to pt (x,y)) is also stupid, but less. I think I see a simple way to best-approximate the four-point egg with (a+b cos t)^2+(c+d cos 2t)^2. --rwg
participants (3)
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Fred lunnon -
R. William Gosper -
Steve Witham