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 points (u_i, v_i) (called foci) 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. If n is odd, the algebraic degree of the curve is 2 n , while if n is even the degree is 2 n − ( n n / 2 ) .