Ovoids and Translation Ovals

An ovoid in a 3‐dimensional projective geometry PG(3, q ) over the field GF( q ), where q is a prime power, is a set of q 2 +1 points no three of which are collinear. Because of their connections with other combinatorial structures ovoids are of interest to mathematicians in a variety of fields; for from an ovoid one can construct an inversive plane [ 3 ], a generalised quadrangle [ 11 ], and if q is even, a translation plane [ 13 ]. In fact the only known finite inversive planes are those arising from ovoids in projective spaces (see [ 2 ]). Moreover, there are only two classes of ovoids known, namely the elliptic quadric and, for q even and not a square, the Tits ovoids; these will be described in the next section. If the field order q is odd, then it was shown by Barlotti and Panella (see [ 3 , 1.4.50]) that the only ovoids are the elliptic quadrics. This paper contains a geometrical characterisation of the two known classes of ovoids.