Three representations of a hyperbolic quadric of P G(3, q) in AG(2, q)

Abstract

We construct three different representations of a hyperbolic quadric of a projective Galois space PG(3,q) in the affine Galois plane AG(2, q). To do this, we use the representation R, or R(U₁, U₂, π, 3) of the projective space P (r, k), over the field k, in the affine plane A(2, k), over the same field k, called also ”Crashing”, cited in the bibliography [1].  Further applications of this representation are the construction of maximal partial line spreads in PG, q even, a geometric proof of the equivalence between the Desargues and the Veblen theorems and a geometric proof of the equivalence between the Pappus-Pascal theorem and the ”Three stars theorem”.  Those results will soon appear.