Improving Frenet's Frame Using Bishop's Frame

Abstract

The main drawback of the Frenet frame is that it is undefined at those points where the curvature is zero. Further- more, in the case of planar curves, the Frenet frame does not agree with the standard framing of curves in the plane. The main drawback of the Bishop frame is that the principle normal vector N is not in it. Our new frame, which we call the Beta frame, combines, on a large set of curves, the best aspects of the Bishop frames and the Frenet frames. It yields a globally defined normal, a globally defined signed curvature, and a globally defined torsion. For planar curves it agrees with the standard framing of curves in the plane.