Re: simple question....very simple

• To: mathgroup at smc.vnet.net
• Subject: [mg23717] Re: simple question....very simple
• From: Hartmut Wolf <hwolf at debis.com>
• Date: Mon, 5 Jun 2000 01:09:12 -0400 (EDT)
• Organization: debis Systemhaus
• References: <ZrWY4.9633\$aa7.59764@nlnews00.chello.com>
• Sender: owner-wri-mathgroup at wolfram.com

B schrieb:
>
> I'm looking for the funktion to describe a oval....
>

Hello Ben,

the question of course is, what do you consider as an oval,
and how do you want to hold it in your hands: parametric,
implicit, ...

I assume here oval as "being shaped like an egg" and such won't
offer neither an ellipse nor a race track (Indycar or whatever).

To stay succinct let's use a helper ...

poly[coefficients_, var_] :=
coefficients.var^Range[0, Length[coefficients] - 1]

... to make a polynom. So look at

With[{n = 6, p = 2},
ParametricPlot[
Evaluate[{poly[Prepend[1/Range[n]^p, 0], Cos[2 Pi t]],
Sin[2 Pi t]/Sqrt[2]}], {t, 0, 1}]]

To me this seems to be pretty egg-shaped. (You may like to
experiment with n, certainly with n -> Infinity the tip may get
too pointed).  Whereas

With[{n = 3, p = 1},
ParametricPlot[
Evaluate[{poly[Prepend[1/Range[n]^p, 0], Cos[2 Pi t]],
Sin[2 Pi t]/Sqrt[2]}], {t, 0, 1}, PlotRange -> All]]

might be considered less well-shaped, although you might have been
served that soft-boiled quite often.

A way to get "the ideal egg" might be to look at the process of
egg formation *), derive an ODE for that and solve. For the moment, I
lack the imagination to do that. What about you MathGroup?

[So taking the 3rd power (p = 3), and n -> Infinity (egg doesn't vary
much with n) looks marvellous, but why?]

-- Hartmut Wolf

---------
*) consulting the experts with eggs -- the birds -- you will see
there are different solutions, e.g. there are special, cone-shaped eggs
as not to roll off the cliff and get lost in the sea.

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