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Given $n$ points $(x_i,y_i)$ $(1\leq i\leq n)$ on the 2D plane, is there a general way to parameterize the points $(X,Y)$ on the curve introduced by the following equation?
$$\sum_{i=1}^{n} d_i^k=c$$ where $d_i$ denotes the euclidean distance between $(x_i,y_i)$ and $(X,Y)$, and $k$ is a non-negative real number, and $c$ is a constant.

It is well known that for $k=2$ (and any value of $n$) the loci of $(X,Y)$ is a circle and can be easily parameterized. Furthermore, for $k=1,n=2$ the loci is an ellipse and can be parameterized as well. However, the solution to those two cases cannot be easily extended and applied for general values of $n$ and $k$.

Any ideas would be appreciated.

Salo
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  • For $k=2$ and any value of $n$, you still get a conic curve with classical parametrization means. – Jean Marie Aug 18 '20 at 09:16
  • Seeing this previous question of yours, I see your interest has shifted from the product to the sum of distances (to the power $k$) :) is there a connection ? – Jean Marie Aug 18 '20 at 09:23
  • Both questions are derived by simplifying a bigger problem after a few steps of approximation. Solving either one can facilitate the solution to the main problem, although a closed-form solution to the previous question is needed for that goal. – Salo Aug 18 '20 at 09:38
  • A good reference for the (important) case $k=1$: https://math.stackexchange.com/q/124333 – Jean Marie Aug 18 '20 at 09:47
  • Yes I had checked that before, but even for $k=1$ the parameterization is problematic. – Salo Aug 18 '20 at 09:50
  • In some cases, one has to switch to polar representation $r=f(\theta)$ instead of parametric ones $x=\phi(t),y=\psi(t)$. Another track that I have found useful is to build a system of differential equations out of the constraint (here for example MA+MB+MC=constant) and solve it numerically by a Runge-Kutta program, with often very accurate results. – Jean Marie Aug 18 '20 at 09:55

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