From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math,alt.math,alt.math.iams,alt.math.undergrad
Subject: Re: HELP!!CURVES AND SURFACES!!! HELP!!!
Date: 1 Jul 1998 19:49:28 GMT

In article <3595A751.C0F3DB87@hotmail.com>,
Martin Gillstedt  <m_gillstedt@hotmail.com> wrote:
>Can anyone give me an example of 2 regular (i.e. the derivative is never
>zero) simple closed curves in the plane, which are infinitely many times
>differentiable everywhere (absolutely NO corners or edges in derivatives
>of ANY order), and who are not similar to each other, such that their
>isoperimetric quotients are equal?
>
>  The isoperimetric quotient is defined as =4*pi*A / l^2, where
>A is the area that the curve bounds, and l is the arc length of the
>curve.

Am I missing something? Draw any curve of the type you want -- a
cardiod perhaps, or the zero locus of some cubic polynomial in x and y,
or whatever. Compute its isoperimetric quotient, which is between 0
and 1. Whatever its value, there is an ellipse with the same quotient
(more circular when the quotient is near 1, more elliptical when near 0).

>  An analogous problem is in space: Does there exist 2 closed surfaces
>in space, which bound a volume, V, and has an area, S, an is infinitely
>many times differentiable without exception, not similar to each other,
>whose isoperimetric quotients are equal?
>  The isoperimetric quotient is here = 36*pi*V^2 / S^3.

Same trick: every surface may be compared to a circular ellipsoid.
Here you can even compare say
	x^2 + 2 y^2 + 3 z^2 = 1
to
	x^2 + y^2 + k z^2 = 1

for some  k, which I leave to you to compute.

The point is that the isoperimetric quotient is but one number; while it
might be a complete invariant for any one-dimensional family of similarity
classes of shapes, the family of all smooth curves (or surfaces) is
infinite-dimensional.

dave