Orbital and rotational geometry relevant for obliquity constraints of stars hosting imaged companions. The observer is looking down from the z-axis. The x–y plane is the plane of the sky, and the star is centered on the origin. The x-axis points north and the y-axis points east. Left: the inclination of the companion’s orbital plane, i _o , intersects the sky along the line of nodes, which is oriented by Ω _o from true north to the ascending node. In this configuration the orbital angular momentum vector ﹩\overrightarrow{{L}_{o}}﹩ points in the hemisphere facing the observer, with ﹩\overrightarrow{{L}_{o}}﹩ pointing toward the observer along the z-axis when i _o = 0°. Middle: the star’s orientation in space determines the inclination of its equatorial plane, i _*, and the position angle of its line of nodes. The sense of the stellar spin sets its rotational angular momentum vector ﹩\overrightarrow{{L}_{* }}﹩ and the longitude of ascending node Ω_*. Right: if only the inclinations i _o and i _* are measured, but not their true (or relative) orientations in the plane of the sky (Ω _o and Ω_*), then each angular momentum vector can fall anywhere along nested cones opening toward or away from the observer. If Ω _o is known (point A ₁ , for example), but not Ω_*, then the obliquity can be as small as ∣i _o –i _*∣ (at B ₁ ), can reach i _o + i _* for prograde orbits (B ₂ ), or can be as large as π − ∣i _o − i _*∣ (B ₃ ) for retrograde orbits.