Slant pseudo-lines in the hyperbolic plane (Q2825794)

The slant geometry is a new kind of derivative hyperbolic geometries, which is introduced by \textit{M. Asayama} et al. [Rev. Mat. Iberoam. 28, No. 2, 371--400 (2012; Zbl 1247.53021)].NEWLINENEWLINENEWLINEThe slant geometry is characterized by a \(\phi\)-slant pseudo-line, which is defined by a curve in the Lorentz-Minkowski model of the hyperbolic plane (or the hyperbolic space.) When \(\phi=\frac{\pi}{2}\) a \(\phi\)-slant pseudo-line is a geodesic, and when \(\phi=0\) a \(\phi\)-slant pseudo-line is a horocycle.NEWLINENEWLINENEWLINEIn this paper, the author shows that the \(\phi\)-slant pseudo-line has another (of course equivalent and) intuitional definition in the Poincaré disk model. From this theorem, we can be convinced again that the slant geometry is a one parameter family of geometries connecting the usual hyperbolic geometry and the horocycle geometry.NEWLINENEWLINENEWLINEWe remark that this paper concerns only a bit of introduction of the slant geometry and all of us have a great chance to obtain new results of the slant geometry.