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 (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 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 , while if n is even the degree is [5]: (Thm. 1.1)
n-ellipses are special cases of spectrahedra.
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 (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 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 , while if n is even the degree is [5]: (Thm. 1.1)
n-ellipses are special cases of spectrahedra.