In geometry, the -ellipse is a generalization of the ellipse allowing more than two foci. -ellipses go by numerous other names, including multifocal ellipse, polyellipse, egglipse, -ellipse, and Tschirnhaus'sche Eikurve (after Ehrenfried Walther von Tschirnhaus). They were first investigated by James Clerk Maxwell in 1846.
Given focal points in a plane, an -ellipse is the locus of points of the plane whose sum of distances to the foci is a constant . In formulas, this is the set
\left\{(x,y)\inR2:
| n | |
| \sum | |
| i=1 |
| 2 | |
| \sqrt{(x-u | |
| i) |
+
| 2} | |
| (y-v | |
| i) |
=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 of foci, the -ellipse is a closed, convex curve.[1] The curve is smooth unless it goes through a focus.
The n-ellipse is in general a subset of the points satisfying a particular algebraic equation. If n is odd, the algebraic degree of the curve is
2n
2n-\binom{n}{n/2}.
n-ellipses are special cases of spectrahedra.