Three Points Threorem

Three points on a curve.
Three points q1, q2, q3 that satisfy q1 + q2 + q3 = 0 on a given curve. How do you find q1 and q2 for given q3 ?

The following theorems give us a method to find these points.

  1. If the given curve γ is invariant under the inversion with respect to the origin, q --> −q, then let γ|| be a parallel translation of γ to −q3. (So, q3 in γ is moved to the origin in γ||)
    1. Then, points {q1,q2} that satisfy q1+q2+q3=0 are the elements of the cross points of γ and γ||.
    2. Inversely, if there is a cross point q1 of γ and γ||, then, there exists q2 which is also a cross point of the same curves and satisfies q1+q2+q3=0.
    See figs. 1, 2 and animations 1, 2.
  2. If curve γ has no symmetry, γ|| must be replaced by γ* that is the image of γ by the map q --> q*=−q−q3.
    See fig. 3 and animations 3, 4.

Read our paper J. Phys. A: Math. Theor. 42 395205 (16pp) 2009 doi: 10.1088/1751-8113/42/39/395205 or arXiv http://arxiv.org/abs/0906.2249 for more generic theorem.


Example of the Three Points Theorem for origin symmetric convex curve.
Fig. 1: Example for a origin symmetric convex curve.
If the curve is a circle, every one knows very well this theorem is true.


Exapme of the Three Points Theorem for eight-shaped curve. Exapme of the Three Points Theorem for eight-shaped curve.
Fig. 2: Example for a eight-shaped curve.


Exapme of the Three Points Theorem for general convex curve.
Fig. 3: Example for a general convex curve.

The curve γ* is the image of γ by the inversion with respect to the point −q3/2.
Alternatively, the curve γ* can be made by the following two steps.

  1. Make γ' that is the inversion of γ with respect to the origin by the map q --> −q
  2. Make γ* = (γ')|| that is the parallel translation of γ' by the map q --> q −q3.
Therefore, if the curve γ is invariant under the inversion, we get γ'=γ and γ*=γ||.

Animations

Animation 1: Three Points Theorem for the figure-eight solution.
Animation for the Three Points Theorem for the figure-eight solution
The red, green and blue circles represent three bodies for the figure-eight solution found by Moore, Chenciner and Montgomery under the Newtonian gravity. The black eight-shaped curve is the orbit for the solution. We take the red circle for q3, then the positions for other two points that satisfy q1+q2+q3=0 is given by the cross points of the original eight (black curve) and its parallel translation (red curve).


Animation 2: Three Points Theorem for an oval.
Animation for the Three Points Theorem for  an oval


Animation 3: Three Points Theorem for an egg-shaped curve.
Animation for the Three Points Theorem for an egg-shaped curve.


Animation 4: Three Points Theorem for a distorted curve.
Animation for the Three Points Theorem for a distorted curve.