Application To Kinematics
By using the foundations given above, one can show that the mapping
is an ordinary or hyperbolic rotation according as
- .
These mappings are projectivities in the inversive ring geometry of coquaternions. The collection of these mappings bears some relation to the Lorentz group since it is also composed of ordinary and hyperbolic rotations. Among the peculiarities of this approach to relativistic kinematic is the anisotropic profile, say as compared to hyperbolic quaternions.
Reluctance to use coquaternions for kinematic models may stem from the (2, 2) signature when spacetime is presumed to have signature (1, 3) or (3, 1). Nevertheless, a transparently relativistic kinematics appears when a point of the counter-sphere is used to represent an inertial frame of reference. Indeed, if, then there is a p = i sinh(a) + r cosh(a) ∈ J such that t ∈ Dp, and an b ∈ R such that t = p exp(bp). Then if u = exp(bp), v = i cosh(a) + r sinh(a), and s = ir, the set {t, u, v, s} is a pan-orthogonal basis stemming from t, and the orthogonalities persist through applications of the ordinary or hyperbolic rotations.
Read more about this topic: Split-quaternion
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