The figure-eight has a bifurcation at \alpha=-0.21. Where \alpha is the power of the potential energy, U=-\alpha^{-1}/r^\alpha.
The bifurcated solution is choreographic, that breaks the left-right symmetry. So it has non-vanishing angular momentum.
The shape change of the orbit by the parameter \alpha.
Solid circles stand for three bodies at an isosceles configuration,
and hollow circles at another isosceles configuration.
The parameter c is the angular momentum,
d is the difference of the base length of the two isosceles triangles,
and T is the period.
The motion of three bodies at \alpha=-0.200.
The motion of three bodies at \alpha=0, namely the log potential.
The motion of three bodies at \alpha=+0.130.
There are 2 self clossing points.