Binary relations as single primitive notions for hyperbolic three-space and the inversive plane

By interpreting J.A. Lester's [9] result on inversive-distance-preserving mappings as an axiomatizability statement, and by using the Liebmann isomorphism betwen the inversive plane and hyperbolic three-space, we point out that hyperbolic three-spaces (and inversive geometry) coordinatized by Euclidean fields can be axiomatizaed with planes (or circles) as variables, by using only the plane-orthogonality (or circle-orthogonality) predicate ⊥_p (or ⊥_c), or by using only the predicate δ′ (or δ), where δ′(p,p′) (or δ(A, B)) is interpreted as 'the distance between the planes and ′p is equal to the length of the segment s whose angle of parallelism is π/4(i. e. Π(s) = π/4)' (or as 'the numerical distance between the disjoint circles A and B has the value g, which corresponds to s via Liebmann's isomorphism').