Heterogeneous random walk in one dimension: Difference between revisions

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{{DISPLAYTITLE:''n''-ellipse}}
 
In [[geometry]], a '''multifocal ellipse''' (also known as '''''n''-ellipse''', '''''k''-ellipse''', '''polyellipse''', '''egglipse''', '''generalized ellipse''', and (in German) '''[[Ehrenfried Walther von Tschirnhaus|Tschirnhaus]]'sche Eikurve''') is a generalization of an [[ellipse]] with multiple foci.
 
More concretely, and given ''n'' points in a plane ([[Focus (geometry)|foci]]), an ''n''-ellipse is the [[Locus (mathematics)|locus]] of all points of the plane whose sum of distances to the ''n'' foci is a constant. The set of points of an ''n''-ellipse is defined as:
 
: <math>\left\{(x, y) \in R^2: \sum_{i=1}^n \sqrt{(x-u_i)^2 + (y-v_i)^2} = d\right\}</math>
 
The 1-ellipse corresponds to the circle. The 2-ellipse corresponds to the classic ellipse. Both are algebraic curves of degree&nbsp;2.
 
== References ==
 
*J.C. Maxwell: "[http://books.google.com/books?id=zfM8AAAAIAAJ&pg=PA35&lpg=PA35 Paper on the Description of Oval Curves], Feb 1846, from "The Scientific Letters and Papers of James Clerk Maxwell: 1846-1862"
*Z.A. Melzak and J.S. Forsyth: "Polyconics 1. polyellipses and optimization". Q. of Appl. Math., pages 239–255, 1977.
*J. Nie, P.A. Parrilo, B. Sturmfels: "[http://math.ucsd.edu/~njw/PUBLICPAPERS/kellipse_imaproc_toappear.pdf Semidefinite representation of the ''k''-ellipse]".
*P.L. Rosin: "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.159.9549 On the Construction of Ovals]"
*P.V. Sahadevan: "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
*J. Sekino: "''n''-Ellipses and the Minimum Distance Sum Problem". American Mathematical Monthly 106 #3 (March 1999), 193–202. MR 2000a:52003; Zbl 986.51040.
* B. Sturmfels: "[http://www.aimath.org/WWN/convexalggeom/AIM.pdf The Geometry of Semidefinite Programming]"
 
[[Category:Conic sections]]
[[Category:Curves]]

Revision as of 11:17, 8 November 2013


In geometry, a multifocal ellipse (also known as n-ellipse, k-ellipse, polyellipse, egglipse, generalized ellipse, and (in German) Tschirnhaus'sche Eikurve) is a generalization of an ellipse with multiple foci.

More concretely, and given n points 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. The set of points of an n-ellipse is defined as:

{(x,y)∈R2:∑i=1n(x−ui)2+(y−vi)2=d}

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

References

  • J.C. Maxwell: "Paper on the Description of Oval Curves, Feb 1846, from "The Scientific Letters and Papers of James Clerk Maxwell: 1846-1862"
  • Z.A. Melzak and J.S. Forsyth: "Polyconics 1. polyellipses and optimization". Q. of Appl. Math., pages 239–255, 1977.
  • J. Nie, P.A. Parrilo, B. Sturmfels: "Semidefinite representation of the k-ellipse".
  • P.L. Rosin: "On the Construction of Ovals"
  • P.V. Sahadevan: "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
  • J. Sekino: "n-Ellipses and the Minimum Distance Sum Problem". American Mathematical Monthly 106 #3 (March 1999), 193–202. MR 2000a:52003; Zbl 986.51040.
  • B. Sturmfels: "The Geometry of Semidefinite Programming"
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