Symmetric Gobos in a finite projective plane

A gobo G in any incidence structure K is a (perhaps degenerate) tactical configuration having the property that no three points in G are collinear and no three lines in G are concurrent. General results are obtained where K is a finite projective plane of order n and G has k points and k lines such that each point (line) lies on r lines (points) of G. Particular attention is called to the contrast between the case r = 1 and the case r ≠ 1 when k = n.