Comparison of hyperbolic metric and triangular ratio metric in a square

TL;DR

This paper compares the hyperbolic metric and the triangular ratio metric within a square, providing a sharp estimate for their ratio involving hyperbolic functions.

Contribution

It introduces a precise estimate relating hyperbolic and triangular ratio metrics in a square, advancing understanding of their geometric relationship.

Findings

Derived a sharp estimate for the ratio of hyperbolic to triangular ratio metrics.

Established bounds involving hyperbolic functions for these metrics.

Enhanced metric comparison techniques in planar geometry.

Abstract

Let $K$ be a square in the plane and $\rho_K(x,y)$ be the hyperbolic distance between $x$, $y\in K$. Denote by $s_K(x,y)$ the triangular ratio metric in $K$; for $x\neq y$ the value of $s_K(x,y)$ equals the ratio of the Euclidean distance $|x-y|$ between $x$, $y\in K$ to the value $\sup_{z\in \partial K}(|x-z|+|z-y|)$. We obtain a sharp estimate for the ratio of $\th (\rho_K(x,y)/2)$ to $s_K(x,y)$.