What is "n-ellipse"

In geometry, the multifocal ellipse (also known as n-ellipse, k-ellipse, polyellipse, egglipse, generalized ellipse, and (in German) Tschirnhaus'sche Eikurve) is a generalization of the ellipse allowing more than two foci.

Specifically, given n points (u, v) in a plane ( foci), an n-ellipse is the locus of all points of the plane whose sum of distances to the n foci is a constant d. The set of points of an n-ellipse is defined as:

$\left\{(x, y) \in R^2: \sum_{i=1}^n \sqrt{(x-u_i)^2 + (y-v_i)^2} = d\right\}.$

The 1-ellipse corresponds to the circle. The 2-ellipse corresponds to the classic ellipse. Both are algebraic curves of degree 2.

For any number of foci, the curves are convex and closed. If n is odd, the algebraic degree of the curve is 2,  while if n is even the degree is $2^k - \binom{k}{k/2}.$ In the n=3 case, the curve is smooth unless it goes through a focus.