= Facing geometry of synchronous rotation
{title2=$\theta-f=\mathrm{constant}$}
View a <circular orbit> along its perpendicular spin axis. Let $f$ be the orbital angle measured from a fixed direction, choosing an arbitrary zero for the circular limit of <true anomaly>, and let $\theta$ be the direction of an outward arrow painted on the body, measured from the same fixed direction. The direction from the body to its companion is $f+\pi$. Thus an inward-facing arrow satisfies
$$
\theta=f+\pi\pmod{2\pi},\qquad \dot\theta=\dot f=n,\qquad n=\frac{2\pi}{T}.
$$
More generally <synchronous rotation> keeps $\theta-f$ constant; its particular value selects which surface direction faces the companion. Over one <orbital period> $T$, both $f$ and the unwrapped $\theta$ increase by $2\pi$. This is the short geometric explanation for one turn per orbit. For a nonspinning body, $\dot\theta=0$ instead, and the companion direction sweeps through all surface longitudes. On an eccentric orbit, mean $1:1$ spin does not force $\theta-f$ to stay exactly constant: <true anomaly> advances nonuniformly, producing the longitudinal component of <libration>.
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