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Checkerboard incircular nets: Laguerre geometry and parametrisation

  • Alexander I. Bobenko
  • Wolfgang K. SchiefEmail author
  • Jan Techter
Original Paper
  • 13 Downloads

Abstract

We present a procedure which allows one to integrate explicitly the class of checkerboard IC-nets which has recently been introduced as a generalisation of incircular (IC) nets. The latter class of privileged congruences of lines in the plane is known to admit a great variety of geometric properties which are also present in the case of checkerboard IC-nets. The parametrisation obtained in this manner is reminiscent of that associated with elliptic billiards. Connections with discrete confocal coordinate systems and the fundamental QRT maps of integrable systems theory are made. The formalism developed in this paper is based on the existence of underlying pencils of conics and quadrics which is exploited in a Laguerre geometric setting.

Keywords

Discrete differential geometry Confocal conics Laguerre geometry Pencils of quadrics Discrete integrable systems 

Mathematics Subject Classification (2010)

MSC 51B15 MSC 37J35 MSC 52Cxx 

Notes

Acknowledgements

We are indebted to Yuri Suris for insightful comments and discussions. This research was supported by the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics”. W.K.S. was also supported by the Australian Research Council (DP1401000851).

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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Institut für MathematikTU BerlinBerlinGermany
  2. 2.School of Mathematics and StatisticsThe University of New South WalesSydneyAustralia

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