Inverse planes with a given collection of common blocks (Q1332432)

An inversive plane \(I\) is a set of points and circles such that each three distinct points belong to a unique circle. For each point \(P\), there is an affine plane whose points are the points of \(I\) not equal to \(P\) and whose lines are the circles through \(P\). The points of a Miquelian inversive plane are the points of an elliptic ovoid in \(PG(3,q)\); the circles are the intersection with planes of \(PG(3,q)\). In a similar way an inversive plane is defined by the Tits ovoid. (The corresponding affine plane is the Luneburg-Tits plane.) It is possible to have different inversive planes which are defined on the same point set such that for some \(P\), the affine planes obtained by deleting \(P\) are identical. If the number of points on a circle is equal to \(q+1\), it is known that there are \((q^ 2-q)/2\) such inversive planes. The author gives an explicit construction of such a set of inversive planes.