In geometry, the n-ellipse is a generalization of the ellipse allowing more than two foci. n-ellipses go by numerous other names, including multifocal ellipse, polyellipse, egglipse, k-ellipse, and Tschirnhaus'sche Eikurve (after Ehrenfried Walther von Tschirnhaus). They were first investigated by James Clerk Maxwell in 1846.
Given n focal points (ui, vi) 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
The n-ellipse is in general a subset of the points satisfying a particular algebraic equation.: Figs. 2 and 4, p. 7 If n is odd, the algebraic degree of the curve is , while if n is even the degree is : (Thm. 1.1)
n-ellipses are special cases of spectrahedra.
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