Investigation of the kinematics of rigid bodies with dual octonions

In this study, the kinematics of rigid bodies are examined using dual octonions. Within this framework, the definitions of the D^7-module and the dual vector are provided through the use of dual numbers. In addition to the fundamental concepts of dual vectors, the Euclidean unit dual sphere S^6 and its subset P^6 are defined. Subsequently, the E. Study mapping between directed lines in R^7 and points on the unit dual sphere S^6 is derived. It is also demonstrated that every element of the set P^6⊆S^6 corresponds to two intersecting perpendicular directed lines in R^7. The angle between two dual vectors in D^7 is defined. After presenting the fundamental concepts of dual octonions, a dual octonion corresponding to a point in R^7 is defined, and a transformation that applies rotation followed by translation along an axis other than the rotation axis is provided using the total conjugate of the dual octonion. A screw operator, which performs rotation around an axis and translation along the same axis, is defined using a unit dual octonion. Through this screw operator, a new operator is introduced that transforms two intersecting perpendicular directed lines in R^7 into another two intersecting perpendicular directed lines. Finally, several examples are provided to illustrate the theorems and results.