Dual octonions and rigid body kinematics

In this paper, we first give the basic information about octonions and present the Euclidean rotation matrix formed by an octonion in seven-dimensional Euclidean space. Next, we define and introduce the 𝔻�7-module and dual vectors using dual numbers. Then, we provide the transformation that maps the points on the unit dual sphere one-to-one with the directed lines in Double-struck capital R7$$ {\mathrm{\mathbb{R}}} circumflex 7 $$. We also define a subset of the unit dual sphere, demonstrating that each element of this subset corresponds to two intersecting perpendicular directed lines in seven-dimensional Euclidean space. Following that, we introduce dual octonions with their basic algebraic properties and examine rigid body (screw) motions in seven-dimensional Euclidean space using dual octonions. Finally, we define an operator and express that this operator transforms two perpendicular intersecting directed lines in seven-dimensional Euclidean space into two perpendicular intersecting directed lines.