n-ellipse

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

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]

Given $n$ focal points $(u i, v i)$ in a plane, an $n$ -ellipse is the locus of points of the plane whose sum of distances to the $n$ foci is a constant $d$ . In formulas, this is the set

\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, and 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 $2^{n}-{\binom {n}{n/2}}.$ ^[5]^{: (Thm. 1.1)}

n-ellipses are special cases of spectrahedra.

References

^ J. Sekino (1999): "n-Ellipses and the Minimum Distance Sum Problem", American Mathematical Monthly 106 #3 (March 1999), 193–202. MR 1682340; Zbl 986.51040.
^ ^a ^b 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. S2CID 17166889. 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.
^ ^a ^b ^c ^d 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

n-ellipse

See also

References

Further reading