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It is known, via algebraic methods, that a flock of a quadratic cone in PGð3; qÞ gives rise to a family of q þ 1 ovals of PGð2; qÞ and similarly that a flock of a cone over a translation oval that is not a conic gives rise to an oval of PGð2; qÞ. 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 T2ðOÞ for O an oval corresponding to a flock of a translation oval cone in PGð3; qÞ, previously constructed alge- braically.

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