Harmonic ovals

As defined by Tits in 1962, an oval in a projective plane is a set of points, no three collinear, such that each point of the oval lies on a unique tangent to the oval, where tangent is meant in the one-point contact sense. In the finite case, this corresponds to the usual definition, but this formulation allows consideration in infinite planes, and it is the latter case that is the focus of this article. The authors focus on ovals in Moufang planes, in which the little Desargues theorem is satisfied, of which the Desarguesian planes and Pappian planes are special cases. In addition, the authors classify Moufang planes as being Fano planes or not, where in a Fano plane the diagonal points of every quadrangle are collinear. The authors provide a substantial, highly useful survey of key results regarding the various sub-classifications of Moufang planes, and important results regarding ovals in those planes. Specifically of interest are symmetric ovals O, which admit at least one line ℓ such that for any pair of points on O not on ℓ, there is an elation with axis ℓ interchanging the point pair. Ovals which are symmetric with respect to at least one tangent line are called translation ovals, while ovals symmetric with respect to all tangent lines are Moufang ovals. Another important point is that, unlike the finite case, the tangent lines to an oval in an infinite plane need not be well-behaved, so characterizations tend to be restricted to ovals whose tangent lines behave like conics. Of most interest here are central ovals, whose tangent lines are all concurrent. The primary new results in the paper deal with harmonic and strongly harmonic ovals. The concept of a harmonic oval was originally developed for ovals in odd order finite projective planes by Ostrom, as a generalization of a result for real conics due to Brianchon; the extension to even order finite planes is due to Cherowitzo, and the authors provide a unified extension to infinite projective planes. They prove that a central oval in a Moufang Fano plane is a Moufang oval if and only if the oval is harmonic. This result settles an open problem of Ostrom, showing that there exist harmonic ovals that are not conics. However, the authors show that a strongly harmonic oval in a Moufang Fano plane must be a conic in a Pappian plane. Additional results address the situation in Moufang planes that are not Fano planes, and provide several characterizations of ovals which satisfy various degenerations of Pascal's theorem.