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It is a well known fact, that ellipses can be defined as $$\{x\in\mathbb{R}^2\ \ |\ \ \|x-A\|+\|x-B\|-\|B-A\|=e\in\mathbb{R}_0^+;\ A,B\in\mathbb{R}^2\}$$

Question:
has the generalization $$\{x\in\mathbb{R}^2\ \ |\ \ \|x-A\|+\|x-B\|+\|x-C\|-(\|B-A\|+\|C-B\|+\|A-C\|)=e\in\mathbb{R}_0^+;\ A,B,C\in\mathbb{R}^2\}$$ ever been studied and, is the representation of such curves in barycentric coordinates known?

The question is motivated by my quest to generalize planar convex hulls to arbitrary complete graphs.

Calculating the curves could be done with a CAS, but that would not bring about any of its non-trivial properties, which is why I am interested in publications.

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    $\begingroup$ Probably you are looking for 3-ellipses. More generally, one can define k-ellipses, see this MSE question: math.stackexchange.com/questions/124333/… $\endgroup$
    – Francesco Polizzi
    Commented Mar 9, 2016 at 6:47
  • $\begingroup$ @FrancescoPolizzi thanks for supplying the name of those curves and for the pointer to MSE; that is already of help for me. $\endgroup$
    – Manfred Weis
    Commented Mar 9, 2016 at 6:52

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