N-ellipse

In geometry, the multifocal ellipse (1982), a generalized ellipse also known as polyellipse (1977),egglipse (1987),n-ellipse (1999),k-ellipse, and Tschirnhaus'sche Eikurve, is a generalization of the ellipse allowing more than two foci. It was also investigated by James Clerk Maxwell in 1846.

Specifically, given n points (u_i, v_i) in a plane (foci), an n-ellipse is the locus of all points of the plane whose sum of distances to the n foci is a constant d. The set of points of an n-ellipse is defined as:

The 1-ellipse corresponds to the circle. The 2-ellipse corresponds to the classic ellipse. Both are algebraic curves of degree 2.

For any number of foci, the curves are convex and closed. If n is odd, the algebraic degree of the curve is ${\displaystyle 2^{k},}$ while if n is even the degree is ${\displaystyle 2^{k}-{\binom {k}{k/2}}.}$ In the n=3 case, the curve is smooth unless it goes through a focus.

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