n-ellipse

In geometry, the multifocal ellipse (1982),[1] a generalized ellipse also known as polyellipse (1977),[2] egglipse (1987),[3] n-ellipse (1999),[4] k-ellipse,[5] and Tschirnhaus'sche Eikurve, is a generalization of the ellipse allowing more than two foci. It was also investigated by James Clerk Maxwell in 1846.[6]

Specifically, given n points (ui, vi) in a plane (foci), an n-ellipse is the locus of all points of the plane whose sum of distances to the n foci is a constant d. The set of points of an n-ellipse is defined as:

The 1-ellipse corresponds to the circle. The 2-ellipse corresponds to the classic ellipse. Both are algebraic curves of degree 2.

For any number of foci, the curves are convex and closed.[1]:(p. 90) If n is odd, the algebraic degree of the curve is while if n is even the degree is [5]:(Thm. 1.1) In the n=3 case, the curve is smooth unless it goes through a focus.[5]:(Fig. 3)

See also

Further reading

References

  1. 1 2 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.
  2. ↑ Z.A. Melzak and J.S. Forsyth (1977): "Polyconics 1. polyellipses and optimization", Q. of Appl. Math., pages 239–255, 1977.
  3. ↑ 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.
  4. ↑ 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.
  5. 1 2 3 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
This article is issued from Wikipedia - version of the 11/5/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.