The main goal of this work is to show how to exactly compute rotation minimizing frames for an at least three times differentiable space curve . Due to their minimal twist,  these frames are preferable over the usual Frenet one in many contexts , such as in motion design, sweep surface modeling, computer visualization, and in geometric considerations as well. We show that it is possible to find the angle between the principal normal and a rotation minimizing vector  . This is done by first solving the problem for spherical curves and then using the concept of osculating spheres to solve the general case. Finally, since Frenet frames and osculating spheres are not well defined at the zeros of the curvature and torsion , respectively, we discuss the behavior of the angle between the principal normal and a rotation minimizing vectornear these zeros. For a curve whose torsion vanishes on intervals or isolated points only, we show that a rotation minimizing frame can be globally defined using our procedure by conveniently choosing some arbitrary constants. On the other hand, we mention that rotation minimizing frames can be well defined via their Frenet frames for analytic curves. Certainly, our approach will be of great importance in those contexts where rotation minimizing frames play a role .
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 field and prove that they are precisely the plane curves .
R
The main goal of this work is to show how to exactly compute
In this work we furnish characterizations of spherical and plane curves using
D
for an at least three times differentiable space curve
A
in many contexts
D
in many contexts
R
We show that it is possible
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
A
for a spherical curve
R
first solving the problem for spherical curves and then using the concept of osculating spheres to solve the general case. Finally, since Frenet frames and osculating spheres are not well defined at the zeros of the
conveniently writing the curvature and torsion for a curve on a sphere. Later, we extend these expressions for the
R
, respectively, we discuss the behavior of
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
R
angle between the principal normal and a rotation minimizing vectornear these zeros. For a curve whose torsion vanishes on intervals or isolated points only, we show that a rotation minimizing frame can be globally defined using our procedure by conveniently choosing some arbitrary constants. On the other hand, we mention that rotation minimizing frames can be well defined via their Frenet frames for analytic curves. Certainly, our approach will be of great importance in those contexts where rotation minimizing frames play a role
unit tangent and a rotation minimizing vector field and prove that they are precisely the plane curves