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]

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

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 $2^{n}$ , while if n is even the degree is $2^{n}-{\binom {n}{n/2}}.$ ^[5]^{: (Thm. 1.1)}

n-ellipses are special cases of spectrahedra.

n-ellipse

Generalized conic

Geometric median

On the Construction of Ovals

The Geometry of Semidefinite Programming