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. They were first investigated by James Clerk Maxwell in 1846.
Given n points in a plane, an n-ellipse is the locus of all 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. 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, while if n is even the degree is.
n-ellipses are special cases of spectrahedra.
