trainor@ucla-cs.UUCP (02/21/86)
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
ARPA: trainor@locus.ucla.edu
UUCP: ...!{sch-loki,silogic,randvax,ihnp4,sdcrdcf,trwspp,ucbvax}!ucla-cs!trainor