Checkerboard Incircular Nets in Space Forms

Alexander I. Bobenko, Carl O. R. Lutz, Jan Techter

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Description


Incircular nets (ic-nets) are two parameter families of lines with the combinatorics of a square grid such that each elementary quad posseses an incircle. These nets are known to be strongly related to conic sections [1]. If one relexes the requirements on the net such that only every second elementary quad contains an incircle, i.e. the configuration of the circles resembles a checkerboard pattern, one gets checkerboard incircular nets (cic-nets). Cic-nets are less rigid then ic-nets and thus admit to be treated in terms of Laguerre geometry [2], i.e. the theory of oriented lines and circles. Using this machinery one can characterise all those cic-nets generalizing ic-nets (confocal cic-nets) and parametrize them via elliptic functions [3].

Planar constant curvature space forms are simply connected Riemannian surfaces with constant Gaussian curvature. By the uniformization theorem, up to conformal equivalence, these are only the Euclidean plane, hyperbolic plane and Riemann sphere. Since in Riemannian surfaces the notion of lenght and angle is well defined, one gets generalizations of lines and circles. By representing all these space forms throught there respective model on the unit $2$-sphere, one can generalize the notion of ic-net, cic-net and Laguerre geometry.

References


  • Arseniy Akopyan and Alexander Bobenko.
    Incircular nets and confocal conics.
    Transactions of the American Mathematical Society, 370(4):2825–2854, 2018.
    arXiv:1602.04637, doi:10.1090/tran/7292.
  • Wilhelm Blaschke.
    Vorlesungen über Differentialgeometrie und Geometrische Grundlagen von Einsteins Relativitätstheorie. III. Differentialgeometrie der Kreise und Kugeln.
    Vol. 29. Grundlehren Math. Wiss., 1929.
  • Alexander I. Bobenko, Wolfgang K. Schief, and Jan Techter.
    Checkerboard incircular nets: Laguerre geometry and parametrisation.
    Geometriae Dedicata, April 2019.
    arXiv:1808.07254, doi:10.1007/s10711-019-00449-x.

Prof. Dr. Alexander I. Bobenko   +

Projects: A01, A02, C01, B02, Z, CaP, II
University: TU Berlin, Institut für Mathematik, MA 881
Address: MA 881
Tel: +49 (30) 314 24655
E-Mail: bobenko[at]math.tu-berlin.de
Website: http://page.math.tu-berlin.de/~bobenko/


Carl O. R. Lutz   +

Projects: A01
University: TU Berlin
E-Mail: clutz[at]math.tu-berlin.de


Jan Techter   +

Projects: A02
University: TU Berlin
E-Mail: techter[at]math.tu-berlin.de