Generating hyperbolical rotation matrix for a given hyperboloid

Abstract

Hyperbolic rotation is hyperbolically the motion of a smooth object on general hyperboloids given by −a1x2+a2y2+a3z2=±λ, λ∈R+. In this paper, we investigate the hyperbolical rotation matrices in order to get the motion of a point about a fixed point or axis on the general hyperboloids by defining the Lorentzian Scalar Product Space Ra1a2a32,1 such that the general hyperboloids are the pseudo-spheres of Ra1a2a32,1. We adapt the Rodrigues, Cayley, and Householder methods to Ra1a2a32,1 and define hyperbolic split quaternions to obtain an hyperbolical rotation matrix.