I’m curious, but no mathematician. Euclidean geometry is the only type for which I've had training (45 years ago). Apologies in advance for misused terms. A line, as I recall, extends indefinitely in opposite directions. It has no definite “origin” or "center". If we view “translational freedom” as the potential for unrestricted displacement, a line can be said to offer both forward and reverse translational freedoms. A ray, by contrast, has a locatable origin (a closed end) and extends indefinitely in one direction. So, a ray has only forward translational freedom. Structurally then, a ray appears to be half of a line. In fact, two rays joined back-to-back (180° angle), create a line. Still, I am of the impression that a ray and a line offer equivalent total freedoms. By this, I include rotational freedoms for which I find a symmetry. A ray, having a single reference direction “away” from its origin, would seem to offer both an absolute forward (say clockwise) and reverse (say counterclockwise) rotations, with respect to that direction. A line however, having no single reference direction, can offer only forward (non-negative) rotation. For example, observers on either side of an object rotating about a line would disagree about its direction being clockwise or counterclockwise. It rotates or it doesn't, but there's no absolute direction. Is it reasonable to assert that a ray offers twice the rotational freedoms of a line?