With this in hand, we can develop a method for finding a periodic
orbit from an approximation. In 1978 Richardson developed a third
order approximation to the families of halo orbits about *L1, L2,
L3*. Since the Richardson approximations satisfy all the
symmetries of the circular restricted three body problem, they all
pierce the plane *y = 0* perpendicularly. Using the piercing
point as an initial guess to an actual periodic orbit, we have

we have a periodic orbit. Thus we wish to develop a scheme such that after such an integration the x-velocity and z-velocity vanish.

Given the trajectory associated with our initial point, phi, such that:

and

Restricting our change in the X vector to:

we find a new guess by solving

This leads to a linear system of four equations and three unknowns, allowing us to use one variable to parameterize the family of periodic orbits.

The following illustration shows a single employment of the scheme to
locate periodic orbits. The red orbit is
created by integrating the seed forward until it intersects the
*y=0* plane (line in this projection). For ease of illustration
the time-reversal symmetric partner of this orbit is drawn as it's
mirror image below the line *y=0*. This orbit is not periodic
because it does not intersect the *y=0* plane perpendicularly.
We employ the method illustrated above to produce a new guess for the
periodic orbit. The orbit corresponding to this new guess is
green. One can see that this orbit intersects
the *y=0* plane nearly normal, indicating that this is a better
approximation to the periodic orbit. This scheme is re-applied until
an approximation to a periodic orbit is found to the desired accuracy.