Many of the fits gravitate towards theta_inf = 0 (parabola). But we also need to fill in the space between the parabola and the circle, which is occupied by fat ellipses, and then go past the circle into thin ellipses.
I think that the whole range can be covered using an eccentricity e, which is positive for ellipses (e=1 for circle), and negative for hyperbolas, with e=0 corresponding to the parabola.
Many of the fits gravitate towards theta_inf = 0 (parabola). But we also need to fill in the space between the parabola and the circle, which is occupied by fat ellipses, and then go past the circle into thin ellipses.
I think that the whole range can be covered using an eccentricity e, which is positive for ellipses (e=1 for circle), and negative for hyperbolas, with e=0 corresponding to the parabola.