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I recently learned about k-ellipses (see this other question) and I wondered if there was such a thing about k-hyperbolae. I have seen the Wikipedia article on 'Generalized Conics', but I'm not able to discern their application to hyperbolae.

The question I have is what would hyperbolae with multiple focal points look like (if they even exist)? A picture of a 3-ellipse is given on Wikipedia's N-ellipse page. I can imagine something akin to a 3-hyperbola being hyperbolas between any two of the vertices, but I'm not sure if that follows the generalized conic description.

An additional question is if they do exist, is there a literature on them?

wa03
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  • Interesting question. You might define your generalization by requiring that the combination $d_1 + d_2 -d_3$ of distances to three foci be constant. You could try this for any number of foci and any combination of $\pm$ signs. – Ethan Bolker Apr 20 '19 at 15:57
  • Yes, I figured it would be something like that. But I wasn’t sure how many sheets there would be. And before I tried to dig too deep. I wanted to know if there was already an existing literature for the concept and I just didn’t know the terminology. – wa03 Apr 22 '19 at 01:42
  • In the plane these will be curves - perhaps weird. My guess is that there's no literature. There is information about weighted N-ellipses, but the weights on the distances to sum are all positive. – Ethan Bolker Apr 22 '19 at 01:48

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I got some $3$-conics for you! I hope it helps with the "what do they look like" bit :)

3-conics

The equation are as suggested in the comments to your question, linear combinations of distances equaling a constant. To be more precise, we have

  • $d_A+d_B+d_C = 4$ (in red)
  • $-2d_A+d_B+d_C = 1$ (in blue)
  • $d_A+d_B-2d_C = 1$ (in violet)
  • $-d_A-d_B+2d_C = 1$ (in yellow)
  • $2d_A-d_B-d_C = 1$ (in green)

I omitted some redundant choices of signals since points $A$ and $B$ are interchangeable. The $2$ factor is present in most of the equations for a balancing purpose. Point $C$ is following a lemniscate path, if your curious.

This (poor quality) animation was obtained through this Geogebra applet I made.

Alma Arjuna
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