Octagonality conditions in projective and affine planes

Abstract

A projective confined configuration ℭ with axis and centre will be introduced in terms of a non-degenerate octagon \(\mathfrak{D}\) satisfying some hypotheses on the position of its diagonal points (i.e. intersections of edges having distance 8 in the flag graph Γ(\(\mathfrak{D}\))) and its first minor diagonal lines (i.e. diagonal lines joining vertices of distance 6 in Γ(\(\mathfrak{D}\))). That confined configuration gives rise to a certain configurational condition whose affine specialization (i.e. the axis coincides with the line at infinity) is equivalent to the affine Pappos condition, whereas its ‘little’ specialization (i.e. the centre lies on the axis) turns out to be equivalent to the little Desargues condition. In Pappian projective planes ℭ can be completed to a configuration of type (124, 163).