n-ellipse

In geometry, the multifocal ellipse (1982),^[1] a generalized ellipse also known as polyellipse (1977),^[2] egglipse (1987),^[3] n-ellipse (1999),^[4] k-ellipse,^[5] and Tschirnhaus'sche Eikurve, is a generalization of the ellipse allowing more than two foci. It was also investigated by James Clerk Maxwell in 1846.^[6]

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:

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

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 ${\displaystyle 2^{k},}$ while if n is even the degree is ${\displaystyle 2^{k}-{\binom {k}{k/2}}.}$^{[5]:(Thm. 1.1)} In the n=3 case, the curve is smooth unless it goes through a focus.^{[5]:(Fig. 3)}

See also

• Generalized conic

Further reading

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

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. Z.A. Melzak and J.S. Forsyth (1977): "Polyconics 1. polyellipses and optimization", Q. of Appl. Math., pages 239–255, 1977.
3. 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. MR 88b:51041; Zbl 613.51030.
4. J. Sekino (1999): "n-Ellipses and the Minimum Distance Sum Problem", American Mathematical Monthly 106 #3 (March 1999), 193–202. MR 2000a:52003; Zbl 986.51040.
5. J. Nie, P.A. Parrilo, B. Sturmfels: "Semidefinite representation of the k-ellipse".
6. 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