Harmonic points and the intersections of ovals (Q1065356)

Let X be a harmonic point with respect to an oval \({\mathcal O}\) of a finite projective plane of odd order [cf. \textit{T. G. Ostrom}, Geometry-von Staudt's point of view, Proc. Nato Adv. Study Inst., Bad Windsheim/Ger. 1980, 175-196 (1981; Zbl 0462.51003)]; let m be a line through X and f a non-identical (X,m)-elation. After proving a few preliminary results, the author shows that the maximal number of non-fixed points (under f) contained in the set \({\mathcal O}\cap f({\mathcal O})\) is 1 if m is tangent to \({\mathcal O}\) resp. 2 if m is secant to \({\mathcal O}\). For an (X,m)-transitive plane this result is then used for the study of the properties concerning the intersections of \({\mathcal O}\) with its images under the (X,m)-elations.