What's both amusing and frustrating is that there is no such metric on, e.g., the configuration space of all affine lines in R^2. (I.e., all loci of ax + by + c = 0 where a and b are not both 0.)
It's fun to verify (and well-known) that topologically, this space 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. The set of affine lines is simply the set of projective lines with the line at infinity removed. So, its configuration space is a projective plane with a single point removed. We know what the real projective plane looks like: http://en.wikipedia.org/wiki/File:ProjectivePlaneAsSquare.svg So, we can just have some topological fun puncturing and deforming it. When we puncture the square (which we have deformed into a disc), we obtain an annulus with antipodal points on the outer circumference identified with each other. Now, suppose we start with a Möbius strip, and tear along it until it splits into a loop of twice its original length. (We imagine that the points at either side of the tear are still 'connected' by 'wormholes', so this is still topologically a Möbius strip.) This loop is an anullus with a smooth edge and a torn edge. Without loss of generality, we define the smooth edge to be the 'inside' of the annulus and the rough edge to be the 'outside'. Observe that tearing the Möbius strip would have split every point along the tear into two antipodal points on the outer circumference of the annulus, which are still connected via wormholes. Hence, the Möbius strip is topologically equivalent to the punctured real projective plane. Sincerely, Adam P. Goucher