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Nets of lines with the combinatorics of the square grid and with touching inscribed conics. (English) Zbl 1477.51002

Summary: In the projective plane, we consider congruences of straight lines with the combinatorics of the square grid and with all elementary quadrilaterals possessing touching inscribed conics. The inscribed conics of two combinatorially neighbouring quadrilaterals have the same touching point on their common edge-line. We suggest that these nets are a natural projective generalisation of incircular nets. It is shown that these nets are planar Koenigs nets. Moreover, we show that general Koenigs nets are characterised by the existence of a \(1\)-parameter family of touching inscribed conics. It is shown that the lines of any grid of quadrilaterals with touching inscribed conics are tangent to a common conic. These grids can be constructed via polygonal chains that are inscribed in conics. The special case of billiards in conics corresponds to incircular nets.

MSC:

51E15 Finite affine and projective planes (geometric aspects)
51E20 Combinatorial structures in finite projective spaces
51A05 General theory of linear incidence geometry and projective geometries
53A20 Projective differential geometry
52C30 Planar arrangements of lines and pseudolines (aspects of discrete geometry)
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