N-ellipse
# N-ellipse

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 $(u_i,$ $v_i)$ 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

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

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 $equation.^{:Figs.2and4,p.7}$ 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.

