A geometrical construction of the oval(s) associated with an α-flock

It is known, via algebraic methods, that a flock of a quadratic cone in PG(3; q) gives rise to a family of q + 1 ovals of PG(2; q) and similarly that a flock of a cone over a translation oval that is not a conic gives rise to an oval of PG(2; q). In this paper we give a geometrical construction of these ovals and provide an elementary geometrical proof of the construction. Further we also give a geometrical construction of a spread of the GQ T-2(O) for O an oval corresponding to a flock of a translation oval cone in PG(3; q), previously constructed algebraically.