Local bundle theorems in Laguerre planes (Q1074862)

A Laguerre plane is an incidence structure \(L=(P,B\cup C,I)\) with P, B and C non-empty disjoint sets which are respectively called the point set, the line set and the circle set of L, \(I\subset P\times (B\cup C)\) the incidence relation satisfying the following axioms: (i) every point of L is incident with exactly one line of B, (ii) every residual plane of L is an affine plane, and (iii) every circle of L is incident with at least one point of L. Two properties, called local bundle theorems, are defined: The bundle theorem SB1 holds on \(\{K_ 1,\{K_ 2,K_ 2\}\}\) if for any point \(x_ i,y_ i\) on the lines \(K_ i\), \(i\in \{1,2,3\}\) the tangency of the circles \(x_ 1x_ 2x_ 3\) and \(x_ 1y_ 2y_ 3\) implies that the circles \(y_ 1x_ 2x_ 3\) and \(y_ 1y_ 2y_ 3\) are tangent. The bundle theorem SB2 holds on \(\{K_ 1,K_ 2\}\), \(\{K_ 3,K_ 4\}\) if for any points \(x_ i,y_ i\) on \(K_ i\), \(i\in \{1,2,3,4\}\) for which \(x_ 1,x_ 2,x_ 3,x_ 4\) are concyclic, the points \(y_ 1,y_ 2,y_ 3,y_ 4\) are concyclic. The author determines the influence of these on the residual planes. It turns out that some of the residual planes are dual translation planes. Analogous results on what are called special Laguerre planes are also obtained.