On the dual quaternion geometry of screw motions

In this study, the screw motions are studied using dual quaternions with the help of di erent perspectives. Firstly, orthogonality definition of dual quaternions is given and geometric interpretation of orthogonality condition is made. Then, the definition of dual circle is given using orthogonal dual quaternions and it is proved that this dual circle can represent the set of all screw motions. Also, these given theorems are reinforced with some conclusions. In addition, it is seen that a dual quaternion represents a screw motions as a screw operator therefore, other dual quaternions derived from the same dual quaternion represent the same screw motions. Then, it is seen that a screw motions symbolized by a dual quaternion transforms one dual vector to another, and when the sign of the dual vectors changes, it provides the same screw motions. Consequently, the answer of the question “Which dual circles symbolizing screw motions are dual orthonormal to each other?” is given and an important conclusion is obtained regarding this.