A projective confined configuration E with axis and centre will be introduced in terms of a non-degenerate octagon O satisfying some hypotheses on the position of its diagonal points (i.e. intersections of edges having distance 8 in the flag graph F(Β£?)) and its first minor diagonal lines (i.e. diagonal lines joining vertices of distance 6 in F(Β£))). 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 ~ can be completed to a configuration of type (124, 163).
INTRODUCTION AND DEFINITIONS
Configurational conditions have always played an important rrle when dealing with foundations of projective and affine geometry. A first reference is given by the well-known correspondence that has been established between planes satisfying the Pappos, Desargues, little Desargues, and affine little Desargues condition and planes coordinatized respectively by fields, skewfields, alternative fields, and quasi-fields (see, e.g., [10]).
Actually, configurational conditions are even more closely related to regularity conditions for some group G of projectivities of a line l o onto itself, like the condition(P,): 'Each projectivity of G with n fixed points is the identity' (cf. [1]). Indeed, confined configurations appear immediately and universally in planes where a group G of projectivities satisfies the condition (P,) for some n s N:Let ~ be an open incidence structure which consists of (i) n distinct fixed points P1 ..... P, e l o of some projectivity g= I- [ [li,~i, li+l] i β’ Z/mZ of G where X i indicates a pencil of lines containing neither li nor I i + l, (ii) all the lines li (i = 0,..., m), (iii) all points and projection lines obtained successively from P1 .... , P, by projection and section according to the perspectivities 7c is composed of, cf., e.g, [1]; then ~ becomes a confined configuration if one adds a further point P,+ 1 and tracks down its projections and sections, since ~ acts as identity by virtue of(P,), i.e. P~+i = P,+i (cf. [12]). This idea finds its concisest expression in the 'if'-part of the fundamental theorem of projective geometry, due to G. K. Ch. von Staudt and nowadays