Generalized pentagonal geometries

A pentagonal geometry PENT( 𝑘,𝑟 ) is a partial linear space, where every line is incident with 𝑘 points, every point is incident with 𝑟 lines, and for each point 𝑥 , there is a line incident with precisely those points that are not collinear with 𝑥 . Here we generalize the concept by allowing the points not collinear with 𝑥 to form the point set of a Steiner system 𝑆(2,𝑘,𝑤) whose blocks are lines of the geometry.