Introduction to Cartan numbers and some geometric applications

Abstract

This article aims to define the rotational motion around a lightlike axis in pseudo‐null or null frames more easily and to put forward a suitable number system to express rotation transformation in these frames. Therefore, for null and pseudo‐null frames, we define a new set of numbers consisting of a linear combination of two lightlike units and a spacelike unit, which we call Cartan numbers. The first part gives the properties of the set of Cartan numbers, and in the second part, some geometric definitions, theorems, and applications are given. In addition, the definition of the center of a parabola and the concept of the central angle are defined. The connection is made between these concepts and the parabolic rotation transformation. Rotation matrices for parabolic rotations are obtained. The rotating points indicated that it follows in which cases a line path and in which cases a parabolic path. The rotation transformation along any straight line or parabola is studied with examples.