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

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

Given n focal points (ui, vi) 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

$\left\{(x,y)\in \mathbf {R} ^{2}:\sum _{i=1}^{n}{\sqrt {(x-u_{i})^{2}+(y-v_{i})^{2}}}=d\right\}.$ 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.: (p. 90)  The curve is smooth unless it goes through a focus.: p.7

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

n-ellipses are special cases of spectrahedra.