n-ellipse

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N-ellipse.svg

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 points (ui, vi) (called foci) in a plane, an n-ellipse is the locus of all 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. 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 If n is odd, the algebraic degree of the curve is , while if n is even the degree is .[5]:(Thm. 1.1)

See also[edit]

References[edit]

  1. ^ J. Sekino (1999): "n-Ellipses and the Minimum Distance Sum Problem", American Mathematical Monthly 106 #3 (March 1999), 193–202. MR 2000a:52003; Zbl 986.51040.
  2. ^ a b Paul Erdős; István Vincze (1982). "On the Approximation of Convex, Closed Plane Curves by Multifocal Ellipses" (PDF). Journal of Applied Probability. 19: 89–96. JSTOR 3213552. Retrieved 22 February 2015. 
  3. ^ Z.A. Melzak and J.S. Forsyth (1977): "Polyconics 1. polyellipses and optimization", Q. of Appl. Math., pages 239–255, 1977.
  4. ^ 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. MR 88b:51041; Zbl 613.51030.
  5. ^ a b c J. Nie, P.A. Parrilo, B. Sturmfels: "Semidefinite representation of the k-ellipse".
  6. ^ 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

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