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Hyperbolic elliptic parabolic disks approximated by half distance bands

Gyula Lakos

Abstract

Hyperbolic elliptic parabolic disks can be described by the inequality $\frac{x^2}{C^2}+2y^2-2y\leq0$ ($0<C<1$) in the unit disk based Beltrami--Cayley--Klein model of the hyperbolic geometry, up to hyperbolic congruences. The hyperbolic elliptic parabolic disks considered above are sort of close to their supporting half distance bands given by the inequalities $\frac{x^2}{C^2}+ y^2-1\leq0$ and $y\geq0$. Here we consider what `close' might mean, and we look for even more precise approximations, in terms of area and circumference.

Hyperbolic elliptic parabolic disks approximated by half distance bands

Abstract

Hyperbolic elliptic parabolic disks can be described by the inequality () in the unit disk based Beltrami--Cayley--Klein model of the hyperbolic geometry, up to hyperbolic congruences. The hyperbolic elliptic parabolic disks considered above are sort of close to their supporting half distance bands given by the inequalities and . Here we consider what `close' might mean, and we look for even more precise approximations, in terms of area and circumference.
Paper Structure (8 sections, 162 equations, 7 figures)

This paper contains 8 sections, 162 equations, 7 figures.

Table of Contents

  1. Background --- hyperbolic geometry
  2. Hyperbolic elliptic parabolas, an introduction
  3. The area
  4. Circumference
  5. Background: The Study--de Sitter plane geometry
  6. Circumference II
  7. Computation in the Beltrami--Poincaré half-plane model
  8. Reflections

Figures (7)

  • Figure : Fig. \ref{['fig:figHEP01']} Geometric elements related the $h$-elliptic parabola [red].
  • Figure : Fig. \ref{['fig:figHEP02']} Notable distances and measuring the $h$-elliptic parabola up. $\rightarrow$
  • Figure :
  • Figure : Fig. \ref{['fig:figHEP03']} To $E^C=B^C_{\mathrm{up}\,\mathop{\mathrm{artanh}}\nolimits C^2/(2-C^2)}+ B^C_{3/5}-A^C$ in hyperbolic area.
  • Figure : Fig. \ref{['fig:figHEP05']} Lineal cut-offs.
  • ...and 2 more figures