TY - JOUR

T1 - Legendre-like theorems in a general absolute geometry.

AU - Pianta, Silvia

AU - Marchi, Mario

AU - Karzel, Helmut

PY - 2007

Y1 - 2007

N2 - In this paper the axiomatic basis will be a general absolute plane
A = (P,L, α,≡) in the sense of [6], where P and L denote respectively the
set of points and the set of lines, α the order structure and ≡ the congruence,
and where furthermore the word “general” means that no claim is made on
any kind of continuity assumptions. Starting from the classification of general
absolute geometries introduced in [5] by means of the notion of congruence,
singular or hyperbolic or elliptic, we get now a complete characterization of
the different possibilities which can occur in a general absolute plane studying
the value of the angle δ defined in any Lambert–Saccheri quadrangle or,
equivalently, the sum of the angles of any triangle. This yelds, in particular, a
Archimedes-free proof of a statement generalizing the classical “first Legendre
theorem” for absolute planes.

AB - In this paper the axiomatic basis will be a general absolute plane
A = (P,L, α,≡) in the sense of [6], where P and L denote respectively the
set of points and the set of lines, α the order structure and ≡ the congruence,
and where furthermore the word “general” means that no claim is made on
any kind of continuity assumptions. Starting from the classification of general
absolute geometries introduced in [5] by means of the notion of congruence,
singular or hyperbolic or elliptic, we get now a complete characterization of
the different possibilities which can occur in a general absolute plane studying
the value of the angle δ defined in any Lambert–Saccheri quadrangle or,
equivalently, the sum of the angles of any triangle. This yelds, in particular, a
Archimedes-free proof of a statement generalizing the classical “first Legendre
theorem” for absolute planes.

KW - Legendre theorem

KW - absolute geometry

KW - classification of absolute geometry

KW - Legendre theorem

KW - absolute geometry

KW - classification of absolute geometry

UR - http://hdl.handle.net/10807/54951

U2 - 10.1007/s00025-007-0258-0

DO - 10.1007/s00025-007-0258-0

M3 - Article

SN - 1422-6383

SP - 61

EP - 71

JO - Results in Mathematics

JF - Results in Mathematics

ER -