Adv. Appl. Cliﬀord Algebras 27 (2017), 2841–2853
2017 Springer International Publishing
published online March 16, 2017
Applied Cliﬀord Algebras
Rotations on a Lightcone in Minkowski
Communicated by RafalAblamowicz
Abstract. In this paper the generalization of the rotations on any light-
cone in Minkowski 3-space R
is given. The rotation motion on the
lightcone is examined by means of a bilinear form and Lorentzian
notions. We use the corresponding Rodrigues and Cayley formulas and
beneﬁt from the hyperbolic split quaternion product to obtain the cor-
responding rotation matrix.
Mathematics Subject Classiﬁcation. 15A63, 15A66, 53A17, 53A35,
53B30, 70B05, 70B10, 70E17.
Keywords. g-Rotation matrix, Hyperbolic split quaternion, Null axis.
It is well-known that the set of all rotations, together with composition of
rotations, forms rotation group. They can be represented by the orthogonal
matrices, called rotation matrices, so that the rotation group corresponds to a
special non-abelian orthogonal matrix group, denoted by SO(3) in 3 dimen-
sional space. The rotation matrices are very important in geometry, kine-
matics, physics, computer graphics, animations, and optimization problems
involving the estimation of rigid body transformations and other disciplines.
There are various ways of ﬁnding a rotation matrix such as using a unit
quaternion, the Rodrigues formula, the Cayley formula. In the Minkowski
space, the causal character of the axis of rotation has a great importance
in terms of determining the rotational motion. The rotation matrices, whose
group is denoted by SO(1, 2) in R
, about a spacelike or a timelike axis
were generated by using these methods [1,2,5,6,8,12–15]. Also, the rotation
matrix about a null axis is studied in  by using the same methods. In R
they were obtained by means of the corresponding decomposition in and
with Cayley formula in .