TY - JOUR

T1 - Binary relations as single primitive notions for hyperbolic three-space and the inversive plane

AU - Pambuccian, Victor

PY - 2000/12/18

Y1 - 2000/12/18

N2 - 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').

AB - 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').

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U2 - 10.1016/S0019-3577(00)80027-0

DO - 10.1016/S0019-3577(00)80027-0

M3 - Article

AN - SCOPUS:0242300993

SN - 0019-3577

VL - 11

SP - 587

EP - 592

JO - Indagationes Mathematicae

JF - Indagationes Mathematicae

IS - 4

ER -