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Nets of Lines with the Combinatorics of the Square Grid and with Touching Inscribed Conics
Discrete & Computational Geometry ( IF 0.8 ) Pub Date : 2021-02-10 , DOI: 10.1007/s00454-021-00277-5
Alexander I. Bobenko , Alexander Y. Fairley

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.



中文翻译:

线网与正方形网格的组合以及带有接触的内接圆锥线

在投影平面中,我们考虑直线与正方形网格的组合以及所有基本四边形都具有接触式内接圆锥的同余。两个组合相邻的四边形的内切圆锥在其公共边线上具有相同的接触点。我们建议这些网是非圆网的自然射影推广。表明这些网是平面的科尼格斯网。此外,我们表明,一般的柯尼希斯网络的特征在于存在一个接触式圆锥曲线的1参数族。结果表明,具有接触的内接圆锥线的任何四边形网格的线均与公共圆锥线相切。这些网格可以通过圆锥形的多边形链构造。圆锥形台球的特殊情况对应于圆形网。

更新日期:2021-02-10
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