Some remarks on the visual angle metric

TL;DR

This paper explores the relationships between the visual angle metric and the triangular ratio metric in convex domains, identifying extremal points and analyzing how quasiconformal maps distort these metrics.

Contribution

It establishes comparability of the visual angle and triangular ratio metrics in convex domains and characterizes extremal points using hyperbolic geometry, also studying distortion under quasiconformal maps.

Findings

Metrics are comparable in convex domains

Extremal points are characterized in half space and ball

Quasiconformal maps distort these metrics in specific ways

Abstract

We show that the visual angle metric and the triangular ratio metric are comparable in convex domains. We also find the extremal points for the visual angle metric in the half space and in the ball by use of a construction based on hyperbolic geometry. Furthermore, we study distortion properties of quasiconformal maps with respect to the triangular ratio metric and the visual angle metric.