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.


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)
Full Text: DOI arXiv


[1] Akopyan, AV; Bobenko, AI, Incircular nets and confocal conics, Trans. Am. Math. Soc., 370, 4, 2825-2854 (2018) · Zbl 1385.51001
[2] Berger, M., Geometry II. Universitext (2009), Berlin: Springer, Berlin
[3] Bobenko, AI; Hoffmann, T.; Springborn, BA, Minimal surfaces from circle patterns: geometry from combinatorics, Ann. Math., 164, 1, 231-264 (2006) · Zbl 1122.53003
[4] Bobenko, A.I., Lutz, C.O.R., Pottmann, H., Techter, J.: Non-Euclidean Laguerre geometry and incircular nets (2020). arXiv:2009.00978 · Zbl 07393591
[5] Bobenko, AI; Schief, WK; Techter, J., Checkerboard incircular nets: Laguerre geometry and parametrisation, Geom. Dedicata, 204, 97-129 (2020) · Zbl 1432.51003
[6] Bobenko, A.I., Suris, Yu.B.: Discrete Differential Geometry. Integrable Structure. Graduate Studies in Mathematics, vol. 98. American Mathematical Society, Providence (2008) · Zbl 1158.53001
[7] Böhm, W., Verwandte Sätze über Kreisvierseitnetze. Arch. Math. (Basel), 21, 326-330 (1970) · Zbl 0201.53402
[8] Casas-Alvero, E., Analytic Projective Geometry. EMS Textbooks in Mathematics (2014), Zürich: European Mathematical Society, Zürich · Zbl 1292.51002
[9] Chasles, M.: Traité des Sections Coniques Faisant Suite au Traité de Géométrie Supérieure. Premiere Partie. Thesaurus Mathematicae, vol. 3. Physica, Würzburg (1962) · Zbl 0103.38103
[10] Darboux, G.: Lecons sur la Théorie Générale des Surfaces et les Applications Géométriques du Calcul Infinitésimal, vol. 2 & 3. Gauthier-Villars & Fils, Paris (1887, 1889) · JFM 53.0659.02
[11] Darboux, G., Principes de Géométrie Analytique (1917), Paris: Gauthier-Villars, Paris · JFM 46.0877.14
[12] Dragović, V.; Radnović, M., Poncelet Porisms and Beyond. Integrable Billiards, Hyperelliptic Jacobians and Pencils of Quadrics. Frontiers in Mathematics. (2011), Basel: Birkhäuser, Basel · Zbl 1225.37001
[13] Glaeser, G.; Stachel, H.; Odehnal, B., The Universe of Conics (2016), Berlin: Springer Spektrum, Berlin · Zbl 1354.51001
[14] Izmestiev, I., Tabachnikov, S.: Ivory’s theorem revisited. J. Integrable Syst. 2(1), # xyx006 (2017) · Zbl 1401.37062
[15] Khesin, B.; Tabachnikov, S., Pseudo-Riemannian geodesics and billiards, Adv. Math., 221, 4, 1364-1396 (2009) · Zbl 1173.37037
[16] Kœnigs, G., Sur les reseaux plans à invariants égaux et les lignes asymptotiques, Comptes Rendus Acad. Sci., 114, 55-57 (1892) · JFM 24.0130.01
[17] Levi, M.; Tabachnikov, S., The Poncelet grid and billiards in ellipses, Am. Math. Mon., 114, 10, 895-908 (2007) · Zbl 1140.51014
[18] Poncelet, J-V, Traité des Propriétés Projectives des Figures (1865), Paris: Gauthier-Villars, Paris
[19] Richter-Gebert, J.: Meditations on Ceva’s theorem. In: The Coxeter Legacy—Reflections and Projections (Toronto 2004), pp. 227-254. American Mathematical Society, Providence (2006) · Zbl 1104.51001
[20] Richter-Gebert, J., Perspectives on Projective Geometry. A Guided Tour Through Real and Complex Geometry (2011), Heidelberg: Springer, Heidelberg · Zbl 1214.51001
[21] Schwartz, RE, The Poncelet grid, Adv. Geom., 7, 2, 157-175 (2007) · Zbl 1123.51027
[22] Tabachnikov, S., Geometry and Billiards. Student Mathematical Library (2005), Providence: American Mathematical Society, Providence · Zbl 1119.37001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.