It's nice that the space of circles in R^2 is, as a metric space, naturally R^2 x (0,oo). For this is an invariant metric, in that every [permutation of the circles in R^2 that is induced by an isometry of R^2] is also an isometry of the configuration space R^2 x (0,oo). 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. But it's both amusing and frustrating that it has no invariant metric! For, the permutations of the affine lines in R^2 that are induced from isometries of R^2 is identical to the isometry group of R^2, which is 3-dimensional. On the other hand, it can be shown that no matter what metric is put on the Moebius band, its isometry group will be at most 1-dimensional. --Dan Adam wrote: << I wrote: << I'd say that since a circle in R^2 is determined by its center and radius, the space of circles is naturally isometric to R^2 x (0,oo).
Indeed, we can consider a circle to be described by the vertex (in three-dimensional space) of a right circular cone erected upon it. . . .