Abstract
Rotation minimizing (RM) vector fields and frames were introduced by Bishop as an alternative to the Frenet frame. They are used in CAGD because they can be defined even when the curvature vanishes. Nevertheless, many other geometric properties have not been studied. In the present paper, RM vector fields along a curve immersed into a Riemannian manifold are studied when the ambient manifold is the Euclidean 3-space, the hyperbolic 3-space, and a Kahler manifold.
Highlights
Rotation minimizing frames (RMFs) were introduced by Bishop [5] as an alternative to the Frenet moving frame along a curve γ in Rn
An RMF along a curve γ = γ(t) in Rn is an orthonormal frame defined by the tangent vector and n − 1 normal vectors Ni, which do not rotate with respect to the tangent, i.e. Ni′(t) is proportional to γ′(t)
We focused on three situations, according to the case when the ambient manifold is the Euclidean space, the hyperbolic space, and a Kahler manifold: 1. For the case of the Euclidean space R3 we will explicitly show the deep relation between RM vector fields and developable surfaces
Summary
Rotation minimizing frames (RMFs) were introduced by Bishop [5] as an alternative to the Frenet moving frame along a curve γ in Rn. An RMF along a curve γ = γ(t) in Rn is an orthonormal frame defined by the tangent vector and n − 1 normal vectors Ni , which do not rotate with respect to the tangent, i.e. Ni′(t) is proportional to γ′(t). RMFs are widely used in computer-aided geometric design (see, e.g., [9]), in order to define a swept surface by sweeping out a profile in planes normal to the curve. As it is pointed out in [10], the Frenet frame may result in a poor choice for motion planning or swept surface constructions, since it incurs unnecessary rotation of the basis vectors in the normal plane.
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