Innovations beyond the Classical Framework in the Dual Space D3 (Withdrawn Publication)

Abstract

Correspondence between points on the dual unit sphere and lines in the Euclidean space R-3 was first expressed in E. Study Maps, which has served as the foundation for numerous studies in the theory of ruled surfaces and kinematics. The enduring legacy of this theorem lies in the correspondence it establishes between the curves on the dual unit sphere and the ruled surfaces in R-3. Yet, it has prompted fixation on the dual unit sphere, leading to the neglect of the broader D-3 and leaving the theory of curves, surface theory and kinematics in D-3 largely unexplored. To fill this gap, the present study introduces ''generalized E. Study Maps'' that proves that for every dual curve in D-3, there exists a corresponding ruled surface in R-3. Furthermore, the study constructs the theory of curves in D-3 via theory of real curves. The study is expected to guide future research on the dual curve theory, dual surface theory and kinematics in D-3, and pave the way for exploring the magical correspondence between D-3 and R-3 from an expanded viewpoint.