2019/11/18 by Alexander I. Bobenko, Bobenko, Alexander I., Alexander Y. Fairley +1 · 1 citation
Chemistry · Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Chemistry and Stereochemistry Studies #Computational Geometry and Mesh Generation #FOS: Mathematics #Mathematics and Applications #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.1911.08477
openalex publication_date 2019/11/18 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
In the projective plane, we consider congruences of straight lines with the\ncombinatorics of the square grid and with all elementary quadrilaterals\npossessing touching inscribed conics. The inscribed conics of two\ncombinatorially neighbouring quadrilaterals have the same touching point on\ntheir common edge-line. We suggest that these nets are a natural projective\ngeneralisation of incircular nets. It is shown that these nets are planar\nKoenigs nets. Moreover, we show that general Koenigs nets are characterised by\nthe existence of a 1-parameter family of touching inscribed conics. It is shown\nthat the lines of any grid of quadrilaterals with touching inscribed conics are\ntangent to a common conic. These grids can be constructed via polygonal chains\nthat are inscribed in conics. The special case of billiards in conics\ncorresponds to incircular nets.\n