Metric bingles and tringles in H_3

TL;DR

This paper introduces the concepts of bingles and tringles in the Berwald-Moor space, exploring their properties, representations, and connections to exponential angles and poly-numbers, with explicit formulas involving complex integrals.

Contribution

It defines reciprocal and relative bingles, shows their relation to exponential angles and the Berwald-Moor metric, and develops a double-exponential representation of poly-numbers.

Findings

Reciprocal bingles are norms in the space of exponential angles.

The metric of the exponential angle space matches the Berwald-Moor metric.

Explicit formulas for relative bingles and tringles involve complex integrals.

Abstract

In the 3-dimensional Berwald-Moor space are bingles and tringles constructed, as additive characteristic objects associated to couples and triples of unit vectors - practically lengths and areas on the unit sphere. In analogy with the spherical angles, we build two types of bingles (reciprocal and relative). It is shown that reciprocal bingles are norms in the space of exponential angles (in the bi-space H^{\flat}, which exponentially define the representation of poly-numbers. It is shown that the metric of this space coincides with the Berwald-Moor metric of the original space. The relative bingles are connected to the elements of the second bi-space (angles, in the space of angles) H^{2\flat} and allow to provide the doble-exponential representation of poly-numbers. The explicit formulas for relative bingles and tringles contain integrals, which cannot be expressed by means of elementary functions.