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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]

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

Given n points (ui, vi) (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

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

The 1-ellipse is the circle. 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 $$2^{n}$$, 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.

Generalized conic

References

J. Sekino (1999): "n-Ellipses and the Minimum Distance Sum Problem", The American Mathematical Monthly 106 #3 (March 1999), 193–202. MR1682340; Zbl 986.51040.
Erdős, Paul; Vincze, István (1982). "On the Approximation of Convex, Closed Plane Curves by Multifocal Ellipses" (PDF). Journal of Applied Probability. 19: 89–96. doi:10.2307/3213552. JSTOR 3213552. Archived from the original (PDF) on 28 September 2016. Retrieved 22 February 2015.
Z.A. Melzak and J.S. Forsyth (1977): "Polyconics 1. polyellipses and optimization", Q. of Appl. Math., pages 239–255, 1977.
P.V. Sahadevan (1987): "The theory of egglipse—a new curve with three focal points", International Journal of Mathematical Education in Science and Technology 18 (1987), 29–39. MR872599; Zbl 613.51030.
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

James Clerk Maxwell (1846): "Paper on the Description of Oval Curves, Feb 1846, from The Scientific Letters and Papers of James Clerk Maxwell: 1846-1862

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