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So a closed curve is called a circle if all its points are at the same distance from a center point.

Similarly, if for each point in a closed curve the distances to two foci always add up to the same amount, it is called an ellipse.

What is a closed curve called if for each point, the sum of the distances to three or more points is constant?


In equations

$\sqrt{(x-x_{c})^{2}+(y-y_{c})^{2}}=r\\$ Circle

$\sqrt{(x-x_{1})^{2}+(y-y_{1})^{2}}+\sqrt{(x-x_{2})^{2}+(y-y_{2})^{2}}=d\\$ Ellipse

$\sqrt{(x-x_{1})^{2}+(y-y_{1})^{2}}+\sqrt{(x-x_{2})^{2}+(y-y_{2})^{2}}+\sqrt{(x-x_{3})^{2}+(y-y_{3})^{2}}=d\\$ ???

$\vdots$


Also, both the circle and the ellipse can be restated in a $y=$ form (like $y=y_{c}\pm\sqrt{r^{2}-(x-x_{c})^{2}}$ for the circle). That is, splitting the respective curve into two functions of the form $y=f(x)$, since for any given x-coordinate, there are at most two points that satisfy the circle/ellipse's equation. As far as I can tell, the kind of closed curve described by these kinds of equations never change from curving clockwise to anti-clockwise, so there also should be at most two points per x-coordinate that satisfy the equation, so it it seems to me that there must be two y=f(x) functions for each such curve as well, but it seems to be beyond my abilities to extract them from their equations. (I managed to get rid of the roots and isolate all the y terms, but the result is an equation of degree 8, and only special cases of equations above degree 4 are solveable.)

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    $\begingroup$ n-ellipse, also known as multifocal ellipse, polyellipse, egglipse, k-ellipse and Tschirnhaus'sche Eikurve . $\endgroup$ Nov 5, 2017 at 17:05
  • $\begingroup$ Thanks. Is there also a term for the curves generated by subtracting some focal distances from the general sum rather than adding them? Say $sqrt((x-a)^2+(y-b)^2)+sqrt((x-c)^2+(y-d)^2)-sqrt((x-e)^2+(y-f)^2)$ (looks like an ellipse with a dent). $\endgroup$
    – joelproko
    Nov 6, 2017 at 20:15
  • $\begingroup$ As an illustration, this link shows the respective curves of $sqrt((x-1)²+(y+1)²)+sqrt((x-1)²+(y-2)²)+sqrt((x+2)²+y²)=d$, $sqrt((x-1)²+(y+1)²)+sqrt((x-1)²+(y-2)²)=d$ and $sqrt((x-1)²+(y+1)²)+sqrt((x-1)²+(y-2)²)-sqrt((x+2)²+y²)=d$: imgur.com/a/9GvwT (all with a value for $d$ to make them go through the same point; in this case $d={7.311102550928, 0.1, 3.705551275464}) $\endgroup$
    – joelproko
    Nov 7, 2017 at 13:11

1 Answer 1

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What is a closed curve called if for each point, the sum of the distances to three or more points is constant?

Essentially achille hui answered your question in their comment. According to Wikipedia there are several names, including $n$-ellipse, multifocal ellipse, polyellipse, egglipse, $k$-ellipse, and Tschirnhaus'sche Eikurve.

I managed to get rid of the roots and isolate all the y terms, but the result is an equation of degree 8, and only special cases of equations above degree 4 are solveable.

Wikipedia further writes that

If $n$ is odd, the algebraic degree of the curve is $2^n$, while if $n$ is even the degree is $2^n - \binom{n}{n/2}$.[5]:(Thm. 1.1)

So a degree 8 for the case of 3 foci is consistent with this. Solvability isn't discussed in the Wikipedia atricle, but I'd be surprised if it were solvable for generic positions of the foci.

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