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My question is related to the Mice problem, with a slight modification.

Question: Three particles, $A$, $B$ and $C$, are situated at the vertices of an equilateral triangle $ABC$ of side length $d$. Starting at $t=0$, each particle moves with a constant speed $v$. $A$ always has its velocity along $AB$, $B$ along $BC$ and $C$ along $CA$. At what time will the particles meet?

Modification: What would be the equation of the curve followed by the bodies? (If it can be found; provide minimal amount of variables to find it if currently available ones don't suffice.)

Additional challenge: Could this entire problem (with the added modification) be generalized for an $n$-sided polygon following a particular path in each case?

My understanding of this: So in the link above, you can see that it is described as a logarithmic spiral but how do we prove this? Moreover it references a pursuit curve, how do we know this?

Xander Henderson
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user220704
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    Does this answer your question? Case of the 'mice problem' for $n=3$ The question is only about the distance covered, but the answer states the equation of the curve. – joriki May 25 '20 at 11:07
  • @joriki I agree that this question and the linked question are very similar. However, this one is, I think, more general, and is not quite a duplicate. In particular, I feel that a good answer to this question would supersede any answer to the linked question. I've voted to leave this question open, in the hopes that it can attract a more general answer. – Xander Henderson May 25 '20 at 14:02
  • @XanderHenderson: Do you mean you didn't vote to close? Or have I missed some innovation in the voting system? :-) – joriki May 25 '20 at 14:34
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    @joriki I came across the question in the review queue, where the options are "close", "leave open", "edit", and "skip". I voted "leave open" (well, actually, I voted "edit" and made some edits, but I wrote my comment before deciding to edit). – Xander Henderson May 25 '20 at 14:55
  • My question is related to a pursuit curve but it asks for a generalisation to an n-sided polygon. – user220704 Jun 02 '20 at 17:56

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