Yes, that's one way to see it. (But it takes as known something very close to the answer.) A direct way is to note that the family of all lines parallel to a given line forms a line itself perpendicular to them. After you rotate such a parallel family 180 degrees you're back to the original line. So the perpendicular lines form a Moebius band, the space of all lines in the plane. --Dan Adam wrote: << I wrote: << . . . It's fun to verify (and well-known) that topologically, this space [of all lines in the plane] is the (open) Moebius band. . . .
Indeed. The projective lines (ax + by + cz = 0), where (a,b,c) is not (0,0,0) and (la,lb,lc) = (a,b,c) form a projective plane P^2, which is a consequence of projective duality. . . .
________________________________________________________________________________________ It goes without saying that .