In this post I wish to give a construction of some nonclassical generalised quadrangles, that is, ones other than and the dual of . These were discussed in the previous post.
The construction I will discuss is known as Payne derivation and is due to Stan Payne. It constructs new GQs from old ones.
First I need to introduce some terminology and discuss the notion of a regular point. Let be a generalised quadrangle of order . Let be points of . We say that is collinear to if there is a line of the GQ containing both and . We usually denote this by and if and are not collinear we write . For a set of points let denote the set of points collinear with each element of . We say that a point is incident with a line if contains . Again we denote this by .
Now suppose that are two points such that is not collinear with . For each line incident with , since is not on there is a unique point on incident with . Since lies on lines, it follows that has size . Moreover, for each we have that is not incident with , otherwise would be a triangle in the geometry. Thus has size . Hence has size at most . Note that so we in fact have that . If for all points not collinear with we say that is a regular point.