*n*-ellipse

__: Generalization of the ellipse to allow more than two foci__

**Short description**

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

- [math]\displaystyle{ \left\{(x, y) \in \mathbf{R}^2: \sum_{i=1}^n \sqrt{(x-u_i)^2 + (y-v_i)^2} = d\right\}. }[/math]

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 [math]\displaystyle{ 2^n }[/math], while if *n* is even the degree is [math]\displaystyle{ 2^n - \binom{n}{n/2}. }[/math]^{[5]}^{:(Thm. 1.1)}

*n*-ellipses are special cases of spectrahedra.

## See also

## References

- ↑ J. Sekino (1999): "
*n*-Ellipses and the Minimum Distance Sum Problem",*American Mathematical Monthly*106 #3 (March 1999), 193–202. MR1682340; Zbl 986.51040. - ↑
^{2.0}^{2.1}Erdős, Paul; Vincze, István (1982). "On the Approximation of Convex, Closed Plane Curves by Multifocal Ellipses".*Journal of Applied Probability***19**: 89–96. doi:10.2307/3213552. http://renyi.mta.hu/~p_erdos/1982-18.pdf. 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. - ↑
^{5.0}^{5.1}^{5.2}^{5.3}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*

## Further reading

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

Original source: https://en.wikipedia.org/wiki/N-ellipse.
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