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We show that a central oval in a Moufang Fano plane is a Moufang oval if and only if the oval is harmonic. This shows, for the first time, that harmonic ovals that are not conics exist. In addition, we show that, if a strongly harmonic oval in a Moufang Fano plane exists, then the plane is Pappian and the oval is a conic and that if a harmonic oval in a Moufang plane that is not a Fano plane exists, then the plane is Pappian and the oval is a conic. We also show local versions of these results, characterising conics by the property of: being harmonic at external points on a fixed secant line, being harmonic at external points on a fixed external line, being harmonic at points on a fixed tangent line and being harmonic at all secant lines on a fixed point of the oval. Finally, we give some related characterisation of conics in terms of degenerations of Pascal's theorem, including the theorem that an oval in a Moufang plane with the four point Pascal property is a conic in a Pappian plane. [ABSTRACT FROM AUTHOR]

The goal of this paper is an analysis of the geometry of billiards in ellipses, based on properties of confocal central conics. The extended sides of the billiards meet at points which are located on confocal ellipses and hyperbolas. They define the associated Poncelet grid. If a billiard is periodic then it closes for any choice of the initial vertex on the ellipse. This gives rise to a continuous variation of billiards which is called billiard motion though it is neither a Euclidean nor a projective motion. The extension of this motion to the associated Poncelet grid leads to new insights and invariants. [ABSTRACT FROM AUTHOR]

This article looks at subconics of order q of PG (2 , q 2) in the Bruck–Bose representation in PG (4 , q) . A unified characterisation is given, namely a normal rational curve N in PG (4 , q) corresponds to an F q -conic of PG (2 , q 2) if and only if N is 2-special - a notion related to how N interacts with the regular 1-spread of the Bruck–Bose representation. [ABSTRACT FROM AUTHOR]

In a number of previous papers, techniques have been developed to construct geometric 3-configurations whose reduced Levi graph has at least one pair of parallel edges, but construction techniques for configurations whose reduced Levi graph is a simple bipartite cubic graph remained elusive. In Using conics to construct geometric 3-configurations, part I: symmetrically generalizing the Pappus configuration (2016), a primarily straightedge-and-compass construction, along with the construction of a certain conic section, was developed to construct configurations whose reduced Levi graph is an edge-labelled digraph with underlying graph $$K_{3,3}$$ . In this paper, we extend that construction to produce a construction technique for certain classes of configurations whose reduced Levi graphs are bipartite simple cubic graphs that contain at least one quadrilateral; in particular, we produce a geometric construction method that produces configurations whose reduced Levi graphs have as their underlying graphs the cyclic fence graphs, which unify even prism graphs and odd Moebius ladders. [ABSTRACT FROM AUTHOR]

The bilinear flocks of Cherowitzo are generalized to a large variety of bilinear flocks of the cone $${\mathcal{C}_q}$$ . Using these ideas, net replacements of certain Hughes-Kleinfeld planes of order q are obtained that construct every André plane in PG(3, q). [ABSTRACT FROM AUTHOR]

Two new circles (denoted by Γ I and Γ E) are shown to be associated with any ellipse. Their analogies with two circles described by Barlotti are described. Two further new circles—denoted by Ω and Γ—are shown to be associated with any general point P of the ellipse. Tight relationships link the circles Ω and Γ with the circle K (previously introduced by the present author), as well as with Monge’s orthoptic circle, with Barlotti’s circles and with the circles Γ I and Γ E. In particular, the circle Ω is orthogonal to Monge’s circle. A new special point of the ellipse (the point T) is described. New properties of Fagnano’s point are described. [ABSTRACT FROM AUTHOR]

Let H be an ellipse with semiaxes a and b ( a > b). Two circles concentric with H, and with radii a − b and a + b, are described, each of them being the locus of the intersections between couples of noteworthy H-related lines (Theorems 1 and 2). Tight, as well as unexpected links among such circles and Monge’s circle are shown (Theorems 4, 5, and 6). A surprising pythagorean relationship involving segments related to the ellipse is shown (Theorem 3). A set of 10 concyclic points is associated with any general point of H (Theorem 9). New properties of Fagnano’s point are described (Theorems 10 through 13). Only elementary facts from trigonometry and analytic geometry are used. [ABSTRACT FROM AUTHOR]

We give a characterisation of the Rosati oval in the regular nearfield plane of dimension 2 over its centre. [ABSTRACT FROM AUTHOR]

In this paper we characterize the family of external lines to a hyperbolic quadric of PG(3, q). [ABSTRACT FROM AUTHOR]

The André/Bruck and Bose representation ([1], [5, 6]) of PG(2, q²) in PG(4, q) is a tool used by many authors in the proof of recent results. In this paper the André/Bruck and Bose representation of conics in Baer subplanes of PG(2, q²) is determined. It is proved that a non-degenerate conic in a Baer subplane of PG(2, q²) is a normal rational curve of order 2, 3, or 4 in the André/Bruck and Bose representation. Moreover the three possibilities (classes) are examined and we classify the conics in each class. [ABSTRACT FROM AUTHOR]

A point P not on a non-degenerate conic C in PG(2, q), q odd, is called internal to C if no tangent line to C contains P, external otherwise. The set of internal points of C is a q (q - 1) 2 -set of type (0 , q - 1 2 , q + 1 2) . In this paper, we classify all q (q - 1) 2 -sets of class [0, m, n] having exactly two kinds of outer points. [ABSTRACT FROM AUTHOR]