Why does the axis follow the half angle rule?

For translations

pos = § vel (where § is the integral)

For rotations

q = § w q/2 (where q is the angular position and w is the angular velocity)

Thus to keep position (q) on LP , velocity (w) must tilt out of Listing's plane by the half angle rule ( gaze points up 30 deg while w tilts back 15 deg ).