By combining the polarity with a Singer-type generation of the PG, as described by Rao, the construction of the matched lines can be simplified. Essentially, a set of orbits for the lines is created, as follows.

Suppose that the PG is interpreted in terms of homogeneous coordinate, that the polarity is defined by the matrix x'Py=0, so that the point x corresponds to the set of points y in the corresponding plane (and vice versa, of course). Suppose further that a Singer collineation of the points x is given by the matrix R, and of the points y is given by the matrix S. If Rx and Sy are to satisfy the same correlation as x and y, then R'PS = P, or S' = P^-1(R')^-1 P^-1.

This means that a polarity can be represented in terms of initial blocks, and it may be possible to construct a pergola by finding traversals through the difference squares. This seems to work for PGs where q is even, but not when q is odd. See the original papers for more details.

The example is continued here.