An axiom system for the line geometry

Possible primitive notions for geometry of spine spaces

“…Let U be a common point of L 1 , L 2 ; suppose that ∼ 2 (L 1 , L 2 ) does not hold. Let L 3 , L 4 be as required in (12). If both L 3 , L 4 pass through U , then L 3 ∼ L 4 , and thus either L 1 , L 2 , L 3 or L 1 , L 2 , L 4 yield a triangle.…”

“…Thus the sets of the form S(a) are distinguished within C. 2For at least 3-dimensional affine spaces and for at least 4-dimensional projective spaces the result is well known as it is a result of a search for an adequate system of primitive notions for line geometry (cf [8,13,12,9,22]…”

On the axiomatics of projective and affine geometry in terms of line intersection

“…If a 1 b 1 c 1 and a 2 b 2 c 2 are both trilaterals (the tripod case is treated dually), lying in different planes π 1 and π 2 intersecting in g, then we choose a point P on g as the vertex of a tripod x 1 x 2 x 3 , where x 3 = g, x 1 lies in π 1 , and x 2 lies in π 2 . If a 1 b 1 c 1 and a 2 b 2 c 2 were both trilaterals lying in the same plane π, then any x 1 , x 2 , x 3 satisfying the intersection conditions of (10) would have to belong to π, and thus could not form a tripod.…”

“…Pieri [15] first noted that projective three-dimensional space can be axiomatized in terms of lines and line-intersections. A simplified axiom system was presented in [7], and two new ones in [17] and [10], by authors apparently unaware of [15] and [7]. Another axiom system was presented in [16,Ch.…”

Coplanarity of lines in projective and polar Grassmann spaces

“…It was Pieri who first proved that projective three-dimensional space can be axiomatized in terms of lines and binary line intersection [27]. His axiom system has been improved in [14] and independently two other systems have been presented in [16,33]. In [30,Ch.…”