Examples of 3-ellipses for three given foci. The progression of the distances is not linear.

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

${\displaystyle \left\{(x,y)\in \mathbf {R} ^{2}:\sum _{i=1}^{n}{\sqrt {(x-u_{i})^{2}+(y-v_{i})^{2}}}=d\right\}.}$

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

n-ellipses are special cases of spectrahedra.

See also

• Generalized conic
• Geometric median

References

Further reading

• P.L. Rosin: "On the Construction of Ovals"
• B. Sturmfels: "The Geometry of Semidefinite Programming", pp. 9–16.