Orbit invariant under reflection about apsidal vectors

[Happiness]

The book argues that since substituting ##\theta## by ##-\theta## leaves the orbit equation (3.34) unchanged, the orbit is therefore invariant under reflection about the apsidal vectors (Fig 3.12).

If substituting ##\theta## by ##-\theta## leaves the orbit equation (3.34) unchanged, then there exists a plane of symmetry (where ##\theta=0##) in the orbit. How does the book reach the conclusion of invariance under reflection about the apsidal vectors?

 

[gtoguy97]

By showing that the orbit is invariant under reflection about the apsidal vectors, it implies that the plane of symmetry is actually the plane of the apsidal vectors. This is not shown in the book.