n-ellipse

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

In geometry, the Template:Mvar-ellipse is a generalization of the ellipse allowing more than two foci. Template:Mvar-ellipses go by numerous other names, including multifocal ellipse, polyellipse, egglipse, Template:Mvar-ellipse, and Tschirnhaus'sche Eikurve (after Ehrenfried Walther von Tschirnhaus). They were first investigated by James Clerk Maxwell in 1846.

Given Template:Mvar focal points Template:Math in a plane, an Template:Mvar-ellipse is the locus of points of the plane whose sum of distances to the Template:Mvar foci is a constant Template:Mvar. In formulas, this is the set

$\{(x,y)\in {𝐑}^2:\sum _{i=1}^n\sqrt{(x-u_i{)}^2+(y-v_i{)}^2}=d\}.$

The 1-ellipse is the circle, and the 2-ellipse is the classic ellipse. Both are algebraic curves of degree 2.

For any number Template:Mvar of foci, the Template:Mvar-ellipse is a closed, convex curve.Template:Rp The curve is smooth unless it goes through a focus.Template:Rp

The n-ellipse is in general a subset of the points satisfying a particular algebraic equation.Template:Rp If n is odd, the algebraic degree of the curve is $2^n$, while if n is even the degree is $2^n-(\genfrac{}{}{0ex}{}{n}{n/2}).$Template:Rp

n-ellipses are special cases of spectrahedra.

See also

• Generalized conic
• Geometric median

Further reading

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