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. Sometimes it's useful to extend this to R^3, since circles with negative radius (corresponding to upside-down cones) are very elegant for proving Casey's theorem and things of that flavour.
For the general equation below to give a circle, it must hold that B^2 + C^2 > 4AD. So the space of coefficients (mod giving the same circle) isn't quite P^3.
Oh, yes, I've included circles of imaginary radius and suchlike. And probably the union of the line at infinity with an arbitrary straight line, since that's also a degenerate circle (conic passing through the circular points). Thanks for noticing that. I often overlook these matters of diagram dependency. Sincerely, Adam P. Goucher