  If κ stands for a different sort of conic, not an ordinary real one, the function dκ obtained by the above procedure behaves on suitably defined regions of the projective plane like a Euclidian distance function or like the distance function of elliptic geometry (this depends on the nature of the conic κ). Thus, depending on whether κ belongs to one or the other of three kinds of conic, the group of collineations that map κ onto itself is structurally identical with one of the three groups of Lobachevskian, Euclidean, or elliptic isometries. Similar results hold for the three-dimensional case, with κ a quadric surface.