n-ellipse

{{Short description|Generalization of the ellipse to allow more than two foci}}

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File:N-ellipse.svg

In geometry, the {{mvar|n}}-ellipse is a generalization of the ellipse allowing more than two foci. {{mvar|n}}-ellipses go by numerous other names, including multifocal ellipse, polyellipse, egglipse, {{mvar|k}}-ellipse, and Tschirnhaus'sche Eikurve (after Ehrenfried Walther von Tschirnhaus). They were first investigated by James Clerk Maxwell in 1846.

Given {{mvar|n}} focal points {{math|(u_i, v_i)}} in a plane, an {{mvar|n}}-ellipse is the locus of points of the plane whose sum of distances to the {{mvar|n}} foci is a constant {{mvar|d}}. In formulas, this is the set

: $\backslash left\backslash \{(x,\; y)\; \backslash in\; \backslash mathbf\{R\}^2:\; \backslash sum\_\{i=1\}^n\; \backslash sqrt\{(x-u\_i)^2\; +\; (y-v\_i)^2\}\; =\; d\backslash right\backslash \}.$

The 1-ellipse is the circle, and the 2-ellipse is the classic ellipse. Both are algebraic curves of degree 2.

For any number {{mvar|n}} of foci, the {{mvar|n}}-ellipse is a closed, convex curve.{{cite journal|first1=Paul|last1=Erdős|author-link1=Paul Erdős|first2=István|last2=Vincze|author-link2=István Vincze (mathematician)|title=On the Approximation of Convex, Closed Plane Curves by Multifocal Ellipses|journal=Journal of Applied Probability|date=1982|volume=19|pages=89–96|jstor=3213552|url=http://renyi.mta.hu/~p_erdos/1982-18.pdf|accessdate=22 February 2015|doi=10.2307/3213552|s2cid=17166889 |archive-url=https://web.archive.org/web/20160928200222/http://renyi.mta.hu/~p_erdos/1982-18.pdf|archive-date=28 September 2016|url-status=dead}}{{rp|(p. 90)}} The curve is smooth unless it goes through a focus.{{rp|p.7}}

The n-ellipse is in general a subset of the points satisfying a particular algebraic equation.{{rp|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 $2^n\; -\; \backslash binom\{n\}\{n/2\}.$J. Nie, P.A. Parrilo, B. Sturmfels: "[http://math.ucsd.edu/~njw/PUBLICPAPERS/kellipse_imaproc_toappear.pdf 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]{{rp|(Thm. 1.1)}}

n-ellipses are special cases of spectrahedra.