Characterization of Spherical and Plane Curves Using Rotation Minimizing Frames
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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.
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