Hyperbolic elliptic parabolic disks approximated by half distance bands

TL;DR

This paper investigates how hyperbolic elliptic parabolic disks can be closely approximated by half distance bands, focusing on precise measures of their area and circumference within hyperbolic geometry.

Contribution

It introduces a detailed analysis of the approximation of hyperbolic elliptic parabolic disks by half distance bands, refining the understanding of their geometric closeness.

Findings

Quantifies the closeness of hyperbolic disks to half distance bands.

Provides formulas for area and circumference approximations.

Enhances geometric understanding in hyperbolic models.

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.