Geometric generalizations of the E. Study maps and dual curve theory in the dual space ${\mathbb{D}}^3$

The correspondence between points on the unit dual sphere and lines in the Euclidean space ${\mathbb{R}}^3$ was first expressed via E. Study maps, which has served as a foundation for numerous studies in the theory of ruled surfaces and kinematics. The significance of this theorem lies in the correspondence it establishes between the curves on the unit dual sphere and the ruled surfaces in the Euclidean space ${\mathbb{R}}^3$. However, it has led to a strong focus on the unit dual sphere, leading to the neglect of the broader ${\mathbb{D}}^3$ and leaving the theory of curves, surface theory, and kinematics in the dual space ${\mathbb{D}}^3$ largely unexplored. To fill this gap, the present study introduced "generalized E. Study maps", which proved that for every dual curve in the dual space ${\mathbb{D}}^3$, there existed a corresponding ruled surface in the Euclidean space ${\mathbb{R}}^3$. Furthermore, the study constructed the theory of curves in the dual space ${\mathbb{D}}^3$ via the theory of real curves. The results were expected to guide future research on the dual curve theory, dual surface theory, and kinematics in the dual space ${\mathbb{D}}^3$, and pave the way for exploring the striking correspondence between the dual space ${\mathbb{D}}^3$ and the Euclidean space ${\mathbb{R}}^3$ from an expanded viewpoint.