In this work we furnish characterizations of spherical and plane curves using rotation minimizing frames. Due to their minimal twist, in many contexts these frames are preferable over the usual Frenet one, such as in motion design, sweep surface modeling, computer visualization, and in geometric considerations as well. Here, we first furnish an alternative proof for the characterization of spherical curves by using osculating spheres described in terms of a rotation minimizing frame. In addition , we show how to find the angle between the principal normal and a rotation minimizing vector for a spherical curve. This is done by conveniently writing the curvature and torsion for a curve on a sphere. Later, we extend these expressions for the curvature and torsion of a generic curve by studying its behavior near an osculating sphere, i.e., we describe them in terms of spherical analogs. Finally, we also address in this work the problem of characterizing those curves whose position vector  , up to a translation, lies on a (moving ) plane spanned by the unit tangent and a rotation minimizing vector fieldand prove that they are precisely the plane curves .
In this work , we study plane and spherical curves in Euclidean and Lorentz-Minkowski 3-spaces by employing rotation minimizing (RM) frames. By conveniently writing the curvature and torsion for a curve on a sphere , we show how to find the angle between the principal normal and an RM vector field for spherical curves. Later, we characterize plane and spherical curves as curves whose position vector lies , up to a translation, on a moving plane spanned by their unit tangent and an RM vector field. Finally, as an application, we characterize Bertrand curves as curves whose so-called natural mates are spherical .
R
we furnish characterizations of spherical and plane curves using rotation minimizing frames. Due to their minimal twist, in many contexts these frames are preferable over the usual Frenet one, such as in motion design, sweep surface modeling, computer visualization, and in geometric considerations as well. Here, we first furnish an alternative proof for the characterization of spherical curves by using osculating spheres described in terms of a rotation minimizing frame. In addition
, we study plane and spherical curves in Euclidean and Lorentz-Minkowski 3-spaces by employing rotation minimizing (RM) frames. By conveniently writing the curvature and torsion for a curve on a sphere
R
a rotation minimizing vector for a spherical curve. This is done by conveniently writing the curvature and torsion for a curve on a sphere.
an RM vector field for spherical curves.
R
extend these expressions for the curvature and torsion of a generic curve by studying its behavior near an osculating sphere, i.e., we describe them in terms of spherical analogs. Finally, we also address in this work the problem of characterizing those curves
characterize plane and spherical curves as curves
A
lies
R
lies on a (moving )
on a moving
R
the
their
R
a rotation minimizing vector fieldand prove that they are precisely the plane curves
an RM vector field. Finally, as an application, we characterize Bertrand curves as curves whose so-called natural mates are spherical