n-ellipse

In geometry, the multifocal ellipse (also known as n-ellipse, k-ellipse, polyellipse, egglipse, generalized ellipse, and (in German) Tschirnhaus'sche Eikurve) is a generalization of the ellipse allowing more than two foci.

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:

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

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.^{[1]:p. 90} If n is odd, the algebraic degree of the curve is $2^k,$ while if n is even the degree is $2^k - \binom{k}{k/2}.$^{[2]:Thm. 1.1} In the n=3 case, the curve is smooth unless it goes through a focus.^{[2]:Fig. 3}

See also

• Generalized conic

References

1. Paul Erdős; István Vincze (1982). "On the Approximation of Convex, Closed Plane Curves by Multifocal Ellipses" (PDF). Journal of Applied Probability 19: 89–96. JSTOR 3213552. Retrieved 22 February 2015.
2. J. Nie, P.A. Parrilo, B. Sturmfels: "Semidefinite representation of the k-ellipse".

• James Clerk Maxwell: "Paper on the Description of Oval Curves, Feb 1846, from The Scientific Letters and Papers of James Clerk Maxwell: 1846-1862
• Z.A. Melzak and J.S. Forsyth: "Polyconics 1. polyellipses and optimization", Q. of Appl. Math., pages 239–255, 1977.
• P.L. Rosin: "On the Construction of Ovals"
• P.V. Sahadevan: "The theory of egglipse—a new curve with three focal points", International Journal of Mathematical Education in Science and Technology 18 (1987), 29–39. MR 88b:51041; Zbl 613.51030
• J. Sekino: "n-Ellipses and the Minimum Distance Sum Problem", American Mathematical Monthly 106 #3 (March 1999), 193–202. MR 2000a:52003; Zbl 986.51040.
• B. Sturmfels: "The Geometry of Semidefinite Programming", pp. 9-16.