:To those only familiar with a projective line over a field or even a division ring, an unexpected feature is that many of the "homogeneous vectors" (a,b) may be omitted from the space, meaning that the familiar feature of the whole space of "vectors" being partitioned into equivalence classes U(a,b) (points of the line) does not apply, and that the space of "vectors" is not a vector space in teh sense of being closed under addition and (ring) scalar multiplication.  Wording in both the lead and in the body that makes this more salient would be an improvement. —