Generalized Quadrangle - Non-classical Examples

Non-classical Examples

• Let O be a hyperoval in with q an even prime power, and embed that projective (desarguesian) plane into . Now consider the incidence structure where the points are all points not in, the lines are those not on, intersecting in a point of O, and the incidence is the natural one. This is a (q-1,q+1)-generalized quadrangle.
• Let q be a prime power (odd or even) and consider a symplectic polarity in . Choose a random point p and define . Let the lines of our incidence structure be all absolute lines not on together with all lines through p which are not on, and let the points be all points of except those in . The incidence is again the natural one. We obtain once again a (q-1,q+1)-generalized quadrangle