n-Ellipses

DEFINITION: Given n fixed points P_i in the plane and a number d > 0, the n-ellipse is the locus of points P in the plane such that the sum of the distances d_i = d(P, P_i) is d.

PROPOSITION: The 4-ellipse is the set of all intersections of two ellipses with foci P₁, P₂ and P₃, P₄ respectively, such that the sum of the lengths of their major axes is d.

Note that additional restrictions are imposed by the fact that the length of the major axis of an ellipse is greater than or equal than its focal distance.

If P is on the 4-ellipse, then d₁ + d₂ + d₃ + d₃ = d. Let p = d₁ + d₂, then P is on the ellipse with foci P₁, P₂ with major axis p, and the same for 3 and 4 with major axis q = d₃ + d₄, so P is in their intersection. Also, p + q = d₁ + d₂ + d₃ + d₄ = d.

Conversely, if P is in the intersection of both ellipses with major axes p and q, then d₁ + d₂ = p and d₃ + d₄ = q. Together, d₁ + d₂ + d₃ + d₄ = p + q = d, so P is on the 4-ellipse.

PROPOSITION: The 3-ellipse is the set of all intersections of an ellipse with foci P₁, P₂ and a circle with center P₃, such that the sum of the length of the ellipse's major axis and the radius of the circle is d.

Note that the radius of a circle is half of its major axis.

PROPOSITION: The 2-ellipse, more commonly known as the ellipse, is the set of all intersections of two circles with centers P₁ and P₂ such that the sum of their radii is d.

PROPOSITION: The 1-ellipse is the circle with center P₁ and radius d.

THE PICTURE: The line at the bottom specifies the distance d. The other blue points above are the fixed points P_i. All blue points can be moved and the locus will be updated. The black dots show points on the locus, for 1000 values of the parameter, (up to 4000 points, since 2 ellipses intersect at up to 4 points). The green traces show the intersecting ellipses. Use the switch button to switch which points define the ellipses. Use the n-ellipse buttons to switch between the different cases. You can use the pause button to verify the distance sum, and to see whether these loci does not have the ellipse's reflection property.

REFERENCES

Eberly, David. Intersection of Ellipses. [This is where I got the formula for the intersections of 2 ellipses. Note that for two intersecting circles, the quartic equation reduces to a quadratic equation.]
J. Nie, P.A. Parrilo, B. Sturmfels. Semidefinite representation of the k-ellipse. [Contains x-and-y formulas for n-ellipses as 2ⁿ-by-2ⁿ determinants. Note that some terms cancel out so the equation may be less than degree 2ⁿ. The 2-ellipse is the regular ellipse and is of degree 2, same as the 1-ellipse, which is the circle. Also, only a small part of the solution set is the n-ellipse.]
Wikipedia. n-ellipse, Cartesian oval, quartic plane curve, etc. [If 2 of the 3 foci of a 3-ellipse coincide, it's a Cartesian oval with m = 2].
J.C. Maxwell. Paper on the Description of Oval Curves. [The Maxwell of electricity and magnetism.]

Colophon: written directly in HTML 4 and ActionScript 3 for Flash 9.