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## Main Question or Discussion Point

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

A ray, by contrast, has a locatable origin (a closed end) and extends indefinitely in one direction. So, a ray has only

Still, I am of the impression that a ray and a line offer equivalent

A ray, having a single reference direction “away” from its origin, would seem to offer both an absolute

A line however, having no single reference direction, can offer only

Is it reasonable to assert that a ray offers twice the rotational freedoms of a line?

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?

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