Heterogeneous random walk in one dimension

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:

$\{(x,y)\in R^2:\sum _{i=1}^n\sqrt{(x-u_i{)}^2+(y-v_i{)}^2}=d\}$

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"