Obtaining triplet from quaternions

Article history: Received: 24 August 2019 Accepted: 10 May 2020 Available Online: 28 January 2021 In this study, we obtain triplets from quaternions. First, we obtain triplets from real quaternions. Then, as an application of this, we obtain dual triplets from the dual quaternions. Quaternions, in many areas, it allows ease in calculations and geometric representation. Quaternions are four dimensions. The triplets are in three dimensions. When we express quaternions with triplets, our study is conducted even easier. Quaternions are very important in the display of rotational movements. Dual quaternions are important in the expression of screw movements. Reducing movements from four dimensions to three dimensions makes our study easier. This simplicity is achieved by obtaining triplets from quaternions.

The set of quaternions is indicated by .
The dual quaternion set is indicated by ̂ [3].
The triplets is in a three-dimensional space. They can be obtained from arbitrary quaternions in fourdimensional space. So, we can make our study easier.

Triplets and real quaternions
The polar form of a quaternion containing complex module and complex argument was expressed by Sangwine and Bihan (see [1] for detailed information) Let's take A. Atasoy, Y. Yaylı / IJOCTA, Vol.11, No.1, pp.109-113 (2021) where = + is a complex number. Simply, = + + triplet is obtain by multiplying on the left by ̂− 1 of ̂.
It is known that Then it can be written that Accordingly, (for detailed information see [1,3,4]). We can write where | | = , | | = . Accordingly, we can express quaternion with = where is a complex number and is a triplet. Reverse the process, let's take + + unit quaternion. Since cos| | = equality, | | = arccos and sin| | = √1 − 2 . Then = + can be writen where = A leaf is determined with a unit vector . This vector makes a angle with the positive direction in the ( , )-plane. This unit vector can be expressed as = ( ) + ( ) and 2 = −1. This is a pure unit quaternion. In this leaf, let triplet be a non-zero. This leaf is spanned by 1 and . We can write in terms of its components along these vectors. Namely, we can write that = = (cos )1 + (sin ) = where be the angle that makes with the positive vector 1 direction in the leaf.

Dual triplets and dual quaternions
Similar to the polar form of real quaternions, we can also express the polar form of dual quaternions.  and 2 = −1. This is a pure unit dual quaternion. In this leaf, let triplet ̂ be a non-zero. This leaf is spanned by 1 and ̂. We can write ̂ in terms of its components along these vectors. Namely, we can write that ̂=̂= (cos ̂) 1 + (sin ̂) ̂=̂̂.
where ̂ be the dual angle that ̂ makes with the positive vector 1 direction in the leaf.
For every unit dual quaternion, we can write that ̂= coŝ+ŝin̂ = cos( + * ) +ŝin( + * ) where is rotation angle and * is translation component about the dual axis . Accordingly, ̂ is a screw operator. Because ̂ makes dual angular displacement about dual vector axis [3].

Conclusion
Here, triplets are obtained from the real quaternions. Then, dual triplets are obtained from the dual quaternions. Thus, it is believed that the quaternions which have an important place in motion geometry in general are made more useful. This will contribute to the understanding of some concepts such as rotation, translation, displacement and screw movement.
We know that a unit real quaternion is rotation operator. Thus, a unit quaternion can be expressed by two rotation operators.
The displacement of a rigid body is screw displacement. This displacement can be made with screw operator. Every ̂ unit dual quaternion is screw operator. A unit dual quaternion ̂ can be expressed as ̂=̂̂, where ̂ is a unit dual complex number and ̂ is a unit dual triplet. Thus, two screw operators can be expressed with a unit dual quaternion.