Abstract
In this paper, we define some new vector fields along a space curve with nonvanishing curvatures in Euclidean 4-space. By using these vector fields we determine some new planes, curves, and ruled hypersurfaces. We show that the determined new planes play the role of the Darboux vector. We also show that, contrary to their definitions, osculating curves of the first kind and rectifying curves in Euclidean 4-space can be considered as space curves whose position vectors always lie in a two-dimensional subspace. Furthermore, we construct developable and nondevelopable ruled hypersurfaces associated with the new vector fields in which the base curve is always a geodesic on the developable one.
Keywords
Vector (molecular biology)
Vector field
Direction vector
Computer science
Mathematics
Artificial intelligence
Biology
Geometry
Subject Areas
Advanced Differential Equations and Dynamical Systems
·Geometry and Topology ·Physical Sciences
Dynamics and Control of Mechanical Systems
·Control and Systems Engineering ·Physical Sciences
Numerical methods for differential equations
·Numerical Analysis ·Physical Sciences