A constructive theory of ordered affine geometry (Q1279729)

The inclusion of some notion of orientation (or ordering) into the axiomatics for Euclidean geometry is necessary (although missing in Euclid's ``Elements''), as we know since Pasch's ``Vorlesungen über neuere Geometrie'' (1882). The author extends his attempts towards a constructive axiomatization of elementary geometry [begun in the author's paper, Ann. Pure Appl. Logic 76, No. 2, 169-200 (1995; Zbl 0836.03034)], including a notion of ``unequally directed lines''. These axioms allow to deduce Pasch's axiom, and to introduce a betweenness relation (or a pair of orders) on each line. Desargues' property (or embeddability into a space of higher dimension) is not discussed.