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Here are more level sets of the potential. The colors here are:
blue = 3.01,
red = 3.06,
green = 3.15,
orange = 3.225,
magenta = 3.33,
pink = 3.535533 (5/rt(2)),
cyan = 4
yellow = 5,
grey = 7,
light grey = 11,
dark grey = 20
The graph on the left is the orbit of three bodies in the plane, to the right is the orbit in the spherical space. The red, green, and blue colors in the orbit to the left represent the paths of the three bodies. In all of these solutions the three bodies share the same path, they differ only by their starting posisions. You can click and hold down the mouse button on the spherical orbit to rotate it.
There is a label down at the bottom in the center labeled, "Orbit Pattern". This is the the 'pattern' of the orbit. These numbers represent which body is in the middle of the three when all three bodies are in a straight line. This pattern can also be seen on the sphere. When the spherical path passes through the xy plan it will cross somewhere between two of the three collision points. The xy plane is very significant in the spherical orbit. It represents the situation when all three bodies are in a straight line. Depending on which set of two collision points the path crosses between will determine this pattern.
There are three collision points. These are the points when any two bodies have the same position in the plane space. On the sphere they are the three colored dots. The red point, b_12, is (1/2, rt(3)/2, 0), the green point, b_23, is (1/2, -rt(3)/2, 0), and the blue point, b_31, is (-1, 0, 0).
Also, on the sphere are seven different colors of points. Each different color marks a level set of the potential function U. Where U = 1/r_12 + 1/r_23 + 1/r_31 and r_12 = rt(|w| - w.b_12), r_23 = rt(|w| - w.b_23), r_31 = rt(|w| - w.b_31). (. is the dot product) The blue set is U = 3.04, red 3.225, green 3.535533 (5/rt(2)), orange 4.5, magenta 9,
The text field at the bottom of the applet labeled, "Orbit Number" will be used to select different orbits. By entering a number into the text field, you can select any of the compiled orbits. Currently there are 345 orbits. Entering anything other than a number or a number out of range will have no effect on the applet.
By clicking on the orbit to the left you can see the corresponding mapped point to the right.