n-ellipse

In geometry, the n-ellipse is a generalization of the ellipse allowing more than two foci.[1] n-ellipses go by numerous other names, including multifocal ellipse,[2] polyellipse,[3] egglipse,[4] k-ellipse,[5] and Tschirnhaus'sche Eikurve (after Ehrenfried Walther von Tschirnhaus). They were first investigated by James Clerk Maxwell in 1846.[6]

Examples of 3-ellipses for three given foci. The progression of the distances is not linear.

Given n focal points (u_i, v_i) in a plane, an n-ellipse is the locus of points of the plane whose sum of distances to the n foci is a constant d. In formulas, this is the set

${\displaystyle \left\{(x,y)\in \mathbf {R} ^{2}:\sum _{i=1}^{n}{\sqrt {(x-u_{i})^{2}+(y-v_{i})^{2}}}=d\right\}.}$

The 1-ellipse is the circle, and the 2-ellipse is the classic ellipse. Both are algebraic curves of degree 2.

For any number n of foci, the n-ellipse is a closed, convex curve.[2]^{: (p. 90)} The curve is smooth unless it goes through a focus.[5]^: p.7

The n-ellipse is in general a subset of the points satisfying a particular algebraic equation.[5]^{: Figs. 2 and 4, p. 7} If n is odd, the algebraic degree of the curve is ${\displaystyle 2^{n}}$, while if n is even the degree is ${\displaystyle 2^{n}-{\binom {n}{n/2}}.}$[5]^{: (Thm. 1.1)}

n-ellipses are special cases of spectrahedra.