A curve is rectifying if it lies on a moving hyperplane orthogonal to its
curvature vector. In this work, we extend the main result of [B.-Y. Chen,
Tamkang J. Math. \textbf{48} (2017) 209--214] to any space dimension: we prove
that rectifying curves are geodesics on the hypersurface of higher dimensional
cones. We later use this association to characterize rectifying curves that are
also slant helices in three-dimensional space as geodesics of circular cones.
In addition, we consider curves that lie on a moving hyperplane normal to (i)
one of the normal vector fields of the Frenet frame and to (ii) a rotation
minimizing vector field along the curve. The former class is characterized in
terms of the constancy of a certain vector field normal to the curve, while the
latter contains spherical and plane curves. Finally, we establish a formal
mapping between rectifying and spherical curves in any dimension.Comment: 12 pages; keywords: Rectifying curve, geodesic, cone, spherical
curve, plane curve, slant heli