Is there a general equation for an n-ellipse?

Question

I'm sorry if this question is too trivial, but even a more thorough search on Google brought me no answers so far. So please, is there a general equation for n-ellipses? Given N points on the cartesian plane, how can one obtain the equation for the corresponding n-ellipse based on their coordinates? In which cases does such equation exist?

I'm particularly interested in drawing a 3-ellipse right now, so a parametric solution for this particular case would be sufficient, however I'm curious about the existence of equations for the more general case. If necessary, a numeric approximation would also solve my problem.

Thanks in advance!

Answer 1

So please, is there a general equation for n-ellipses ? $~[~\ldots~]~$ I'm curious about the existence of equations for the more general case.

Yes. Usually, it is an implicit polynomial equation of order $2^n$.

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Given n points on the Cartesian plane, how can one obtain the equation for the corresponding n-ellipse based on their coordinates ?

In a similar manner to the following example, detailing the case $n=3$.

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I'm particularly interested in drawing a $3$-ellipse right now, so a parametric solution for this particular case would be sufficient.

• Let the three foci be $(A,B),~(a,b)$, and $(\alpha,\beta).~$ Then we have

$$\sqrt{(x-A)^2+(y-B)^2}+\sqrt{(x-a)^2+(y-b)^2}+\sqrt{(x-\alpha)^2+(y-\beta)^2}=C^2>0$$

• Now employ the following process:

$$\begin{align} \sqrt U+\sqrt V+\sqrt W=C^2\qquad&=>\qquad\sqrt U+\sqrt V=C^2-\sqrt W\qquad\qquad=> \\\\ U+V+2\sqrt{UV}=C^4+W-2C^2\sqrt W\qquad&=>\qquad2\sqrt{UV}+2C^2\sqrt W=C^4+W-U-V \\\\ 4UV+4C^4W+8C^2\sqrt{UVW}=K^2\qquad&=>\qquad8C^2\sqrt{UVW}=K^2-4UV-4C^4W \end{align}$$

$$64C^4UVW=T^2$$

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In which cases does such equation exist ?

It is obvious from the above equations that $K>0$ and $T>0$ are $2\cdot3=6$ necessary conditions $($because $\sqrt W$ can be either one of the three initial radicals, since the equation is symmetrical$)$.

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Thanks in advance !

You're welcome ! :-$)$

Answer 2

Here is an attempt, not totally sure it will work...

You can try to "prescribe" points to be on an ellipse by saying they should be lie on an outer product tensor

$$v^T(T -vv^T)v = 0$$

for each such vector $v$. Then the problem becomes to make T as close as possible so the equations are fulfilled.

We may also need to impose symmetry:

$$T_{i,j} = T_{j,i}$$

And maybe some regularity:

$$\min|T|_2^2$$ or $$\min|T-I|_2^2$$