No abstract

Abstract. The triangular ratio metric is studied in subdomains of the complex plane and Euclidean n-space. Various inequalities are proven for this metric. The main results deal with the behavior of this metric under quasiconformal maps. We also study the smoothness of metric disks with small radii.

An ancient optics problem of Ptolemy, studied later by Alhazen, is discussed. This problem deals with reflection of light in spherical mirrors. Mathematically this reduces to the solution of a quartic equation, which we solve and analyze using a symbolic computation software. Similar problems have been recently studied in connection with ray-tracing, catadioptric optics, scattering of electromagnetic waves, and mathematical billiards, but we were led to this problem in our study of the so-called triangular ratio metric.2010 Mathematics Subject Classification. 30C20, 30C15, 51M99.

Let G R n be a domain and let d1 and d2 be two metrics on G. We compare the geometries defined by the two metrics to each other for several pairs of metrics. The metrics we study include the distance ratio metric, the triangular ratio metric and the visual angle metric. Finally we apply our results to study Lipschitz maps with respect to these metrics.File: hvz20160525arxiv.tex, printed: 2018-10-24, 9.09 2010 Mathematics Subject Classification. 51M10, 30C65.

Abstract. The connection between several hyperbolic type metrics is studied in subdomains of the Euclidean space. In particular, a new metric is introduced and compared to the distance ratio metric.