The transformation in the P-Module by hyperbolic numbers

Abstract

Hyperbolic numbers are one of the well-known number systems, like complex numbers. In this article, we will talk about some properties of hyperbolic numbers, define the P−modulus, and give some properties of the P−modulus. We will also map the points on the unit hyperbolic sphere in the P−module to some special directional lines in E3. Then, we will show the relation between the angle between these directional lines and the hyperbolic angle between the unit hyperbolic vectors.