Seminversive Planes

Finite Laguerre Near-planes of Odd Order Admitting Desarguesian Derivations

“…Olanda [8] considered finite seminversive planes, that is, incidence structures (P, C) with at least two circles and at least three points on each circle such that axiom (J) is satisfied and such that for every circle C ∈ C and any two points p, q, where p ∈ C and q / ∈ C, there are precisely one or two circles passing through q which intersect C only at p. He showed that such a seminiversive plane of order n > 5 is either a Möbius plane of order n or a Möbius plane of order n with one point deleted. However, the last condition in the definition of a seminversive plane is not necessarily satisfied in a Möbius near-plane and so a Möbius nearplane may not be a seminversive plane.…”

“…However, the last condition in the definition of a seminversive plane is not necessarily satisfied in a Möbius near-plane and so a Möbius nearplane may not be a seminversive plane. Thus, Olanda's result [8] cannot be directly applied. However, of course, in the end the same incidence structures are obtained.…”