In geometry, the -ellipse is a generalization of the ellipse allowing more than two foci. -ellipses go by numerous other names, including multifocal ellipse, polyellipse, egglipse, -ellipse, and Tschirnhaus'sche Eikurve (after Ehrenfried Walther von Tschirnhaus). They were first investigated by James Clerk Maxwell in 1846.
Given focal points in a plane, an -ellipse is the locus of points of the plane whose sum of distances to the foci is a constant . In formulas, this is the set
The 1-ellipse is the circle, and the 2-ellipse is the classic ellipse. Both are algebraic curves of degree 2.
For any number of foci, the -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 J. Nie, P.A. Parrilo, B. Sturmfels: "J. Nie, P. Parrilo, B.St.: "Semidefinite representation of the k-ellipse", in Algorithms in Algebraic Geometry, I.M.A. Volumes in Mathematics and its Applications, 146, Springer, New York, 2008, pp. 117-132
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.