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 $(u_{i},&thinsp;v_{i})$ 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. The curve is smooth unless it goes through a focus.

The n-ellipse is in general a subset of the points satisfying a particular algebraic equation. If n is odd, the algebraic degree of the curve is $$2^n$$, while if n is even the degree is $$2^n - \binom{n}{n/2}.$$

n-ellipses are special cases of spectrahedra.