Turning polar angles of a spherical geodesic ($\max |\theta -\pi /2|=\arccos |ah|$)

With polar angle $0\le \theta \le \pi$ and $0<|ah|\le 1$, the spherical-geodesic first integral gives ${\theta }_{\min }=\arcsin |ah|$ and ${\theta }_{\max }=\pi -\arcsin |ah|$. The maximum absolute latitude is the different angle $\arccos |ah|$. At $|ah|=1$ the great circle is the equator; at $h=0$ it is a meridian reaching both poles.