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Focal points $ A,B $ are at distance $ 2 c $ apart. Find locus of a point $P$ so that sum of reciprocals of $AP,BP$ is a constant $ 2/a, a $ is the harmonic mean of these segments. Resemblance is to Cassinian ovals where their product is constant.

$$ \frac{1}{\sqrt{ (x-c)^2 + y^2}} + \frac{1}{\sqrt{ (x+c)^2 + y^2} }= \frac2a $$

Cassinian like Ovals

Narasimham
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    see (https://www.quora.com/What-is-the-graph-of-the-mathematical-curve-when-the-sum-of-the-inverse-distances-from-2-points-is-kept-constant) – Jean Marie Sep 21 '16 at 17:23
  • Ok, so no historical names. – Narasimham Sep 21 '16 at 17:29
  • It looks so. Why not take the opportunity to name them Narasimham curves ? – Jean Marie Sep 21 '16 at 17:31
  • :) .. I am putting a related question about optic/geometrical properties. ..http://math.stackexchange.com/questions/1935374/conics-definition-from-lens-formula – Narasimham Sep 21 '16 at 17:55
  • It is a spiric section: https://en.wikipedia.org/wiki/Spiric_section – Intelligenti pauca Sep 22 '16 at 11:46
  • @Aretino: As mentioned in the Quora link given above, the polynomial representation of the relation has degree 8. Spiric sections only have degree 4. – Blue Sep 22 '16 at 19:10
  • @Blue Exceptional level curve that makes Figure of 8 making saddle point at center has degree eight. Rest of all level curves have a degree four. – Narasimham Sep 22 '16 at 20:44
  • @Narasimham: Interesting. Well, I'll leave my comment up, in case others have are in need of this clarification. – Blue Sep 22 '16 at 20:48
  • @ Jean Marie: I do not know whether any physics of gravity like forces is involved here to be considered of scientific interest. ( Btw, Cassini thought the original ovals named after him were planetary orbits, later disproved by Newton etal.) – Narasimham May 01 '23 at 15:23

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