Mutually diagonal nets on quadrics

Arseniy V. Akopyan, Alexander I. Bobenko, Wolfgang K. Schief, Jan Techter



Canonical parametrisations of classical confocal coordinate systems are introduced and exploited to construct non-planar analogues of incircular (IC) nets on individual quadrics and systems of confocal quadrics. Intimate connections with classical deformations of quadrics which are isometric along asymptotic lines and circular cross-sections of quadrics are revealed. The existence of octahedral webs of surfaces of Blaschke type generated by asymptotic and characteristic lines which are diagonally related to lines of curvature is proven theoretically and established constructively. Appropriate samplings (grids) of these webs lead to three- dimensional extensions of non-planar IC nets. Three-dimensional octahedral grids composed of planes and spatially extending (checkerboard) IC-nets are shown to arise in connection with systems of confocal quadrics in Minkowski space. In this context, the Laguerre geometric notion of inconical octahedral grids of planes is introduced. The latter generalise the octahedral grids derived from systems of confocal quadrics in Minkowski space. An explicit construction of inconical octahedral grids is presented. The results are accompanied by various illustrations which are based on the explicit formulae provided by the theory.


  • Arseniy V. Akopyan, Alexander I. Bobenko, Wolfgang K. Schief, and Jan Techter.
    On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs.
    Discrete and Computational Geometry, August 2020.
    arXiv:1908.00856, doi:10.1007/s00454-020-00240-w.

Prof. Dr. Alexander I. Bobenko   +

Projects: A01, A02, B02, C01, CaP, Z
University: TU Berlin, Institut für Mathematik, MA 881
Address: Straße des 17. Juni 136, 10623 Berlin, GERMANY
Tel: +49 30 31424655
E-Mail: bobenko[at]

Dr. Jan Techter   +

Projects: A02
University: TU Berlin
E-Mail: techter[at]