  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 [Chen 2017,
Tamkang J. Math. 48, 209] to any space dimension: we prove that rectifying
curves are geodesics on hypercones. 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 curves in an
$(m + 2)$-dimensional space and spherical curves in an $(m + 1)$-dimensional
space. A curve is rectifying if it lies on a moving hyperplane orthogonal to
its curvature vector.
