Visual angle metric in the upper half plane

TL;DR

The work addresses how the visual angle metric $v_{\mathbb{H}^2}$, which is not Möbius invariant, relates quantitatively to the hyperbolic metric $\rho_{\mathbb{H}^2}$ on the upper half-plane. It provides an explicit formula for $v_{\mathbb{H}^2}$ in terms of $\rho_{\mathbb{H}^2}$ for normalized pairs in $S^1(0,1)\cap\mathbb{H}^2$, namely $\sin(v_{\mathbb{H}^2}(a,b))=T$ with $T=\frac{|a-b|}{|ab-1|}\left(\frac{1}{2}|a+b|+\sqrt{\Im(a)\Im(b)}\right)$ and $t=\frac{|a-b|}{|ab-1|}=\tanh\big(\rho_{\mathbb{H}^2}(a,b)/2\big)$. The proof blends geometric arguments with computer algebra to obtain explicit intersection formulas and a key collinearity result, enabling the main formula. A second major contribution is a sharp Hölder continuity bound for $K$-quasiregular mappings with respect to $v_{\mathbb{H}^2}$, showing how the visual angle metric distorts under quasiregular maps via a Schwarz-type control. Together, these results enhance the toolkit for comparing the visual angle metric to hyperbolic geometry and for understanding quasiregular distortion in non-Möbius-invariant contexts.

Abstract

We prove an identity which connects the visual angle metric $v_{\mathbb{H}^2}$ and the hyperbolic metric $ρ_{\mathbb{H}^2}$ of the upper half plane $\mathbb{H}^2$. The proof is based on geometric arguments and uses computer algebra methods for formula manipulation. We also prove a sharp Hölder continuity result for quasiregular mappings with respect to the visual angle metric.

Figures (7)

• Figure 1: The hyperbolic disk $B_{\rho_{\mathbb{H}^2}}(m,M)$ centered at the hyperbolic midpoint $m$ of $a$ and $b$ and with the radius $M= \rho_{\mathbb{H}^2}(a,b)/2.$ Now, $v_{\mathbb{H}^2}(a_1,b_1)= \measuredangle(a_1,0,b_1)=2 v_{\mathbb{H}^2}(a,b) \,,\quad {\rm and}\,\, \quad \rho_{\mathbb{H}^2}(a_1,b_1)=\rho_{\mathbb{H}^2}(a,b)\,.$
• Figure 2: The points $a,b$ on the unit circle $S^1$ (on solid line), their hyperbolic middle point $m$, the hyperbolic circle $S_\rho(m,\rho_{\mathbb{H}^2}(a,b)/2)$ (on solid line), and its Euclidean center point $Z$. The points $z_1,z_2$ are the intersection points of the hyperbolic circle $S_\rho(m,\rho_{\mathbb{H}^2}(a,b)/2)$ and the Euclidean circle $S^1({\rm Re}(m),{\rm Im}(m))$ (on dashed line). The supremum $\sup\{\measuredangle(a,x,b)\text{ : }x\in\mathbb{R}^1\}$ is attained at the point $d.$
• Figure 3: The points $p$ and $q$ are the centers of the two circles through $a$ and $b,$ tangent to the real axis at $d$ and $2c-d,$ resp. The points $p$ and $q$ are also the points of intersection of the two parabola with the real axis as the directrix and with foci $a$ and $b,$ resp. Because ${\rm Im}(p)<{\rm Im}(q)$, we see that $v_{\mathbb{H}^2}(a,b)= \max\{ \measuredangle(a,d,b), \measuredangle(a,2c-d,b)\}= \measuredangle(a,d,b)\,.$ The point $m$ here is the hyperbolic midpoint of $a$ and $b.$
• Figure 4: Here $u=LIS[a, \overline{b}, b, \overline{a}]$ and the dashed circular arc is a subarc of the circle centered at the point $c=LIS[a,b,0,1],$ orthogonal to the circle through $a,\overline{a},$ and $b.$ Moreover, $v_{\mathbb{H}^2}(a,b)= \measuredangle(a,d,b).$
• Figure 5: The points $u,s=LIS[a, b_{\ast}, b, a_{\ast}],m,$ and $v$ are collinear. The point $s$ is also the point of intersection of the angle bisectors of the triangle $\triangle(a,b,u).$

Theorems & Definitions (59)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.14
• proof
• Lemma 2.15
• proof
• Proposition 2.16
• proof
• Lemma 2.28
• proof