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From: trainor@ucla-cs.UUCP
Newsgroups: net.math
Subject: With Hanging Ribbons
Message-ID: <9234@ucla-cs.ARPA>
Date: Fri, 21-Feb-86 09:22:19 EST
Article-I.D.: ucla-cs.9234
Posted: Fri Feb 21 09:22:19 1986
Date-Received: Mon, 24-Feb-86 06:43:40 EST
Organization: UCLA Computer Science Department
Lines: 26

Does anyone know of any references to work on lemniscate curves
with an arbitrary number of foci.  Imagine n foci points in the
      _   _        _            _
plane F , F , ..., F .  A point M moves along the curve such that
       1   2        n

the product of its distances from each foci is constant.

   n
-------
 |   |      _  _
 |   | dist(M, F )   =   k
 |   |          i
 i = 1

What about choosing the n foci and constant k such that you come
arbitrarily close to a given non-self-intersecting closed curve.
This was briefly mentioned in this little green book _Remarkable
Curves_ by A.I. Markushevich (translated from Russian), in which
he says, "...proved rigorously in higher mathematics, and the proof
is very complicated."

	Douglas

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