Timelike quaternion frame of a non-lightlike curve (Q945951)

Rotations in the Euclidean 3-space can be expressed using the unit quaternions and corresponding orthogonal matrices can be written in quaternion variables. By identifying columns of an orthogonal matrix with Frenet vectors, Frenet's formulas can be also written in quaternion variables. A split quaternion algebra is an associative, non-commutative non-division ring with four generators \(\{ 1,i,j,k\}\) satisfying the relations \(i^2=-1, j^2=k^2=ijk=1\). The product of two split quaternions can be written in terms of the Lorentzian inner product and the Lorentzian vector product. The set of time-like split quaternions with the operation of split quaternion product form a group which is a 2-fold covering of \(\text{SO}(1,2)\). Rotations in the Minkowski 3-space can be expressed using unit split quaternions. In the paper, the authors reformulate Frenet formulas and parallel transport frame equations in terms of time-like split quaternions.