Abstract
In this paper, we study a weaker version of classic slant helices in Euclidean space R 3 role=presentation> ℝ 3 R 3 \mathbb{R}^3 or Minkowski space R 1 3 role=presentation> ℝ 3 1 R 1 3 \mathbb{R}^3_1 , which will be called general slant helices. We show that any classic slant helix is a general slant helix but the converse is not true. We also obtain equations involving the curvature and torsion that characterize this family of curves.
Highlights
It is usual practice to extend the concepts defined in Euclidean space to other more general spaces, either allowing the metric to be indefinite or allowing the curvature to be nonzero
Most of the time, when the extended concept is applied over the Euclidean space, the original definition is recovered, but other times this is not the case
The term “slant helix” was introduced by Izumiya and Takeuchi [7]: slant helices are defined by the property that their principal normals make a constant angle with a fixed direction
Summary
It is usual practice to extend the concepts defined in Euclidean space to other more general spaces, either allowing the metric to be indefinite or allowing the curvature to be nonzero. The characterization of slant helices given in [7] was extended to the Minkowski space R31 by Ali and López [1] In this case, a regular curve in R31 is said to be a slant helix if there exists a nonzero constant vector v ∈ R31 such that the function ⟨N, v⟩ is constant along the curve, N being the principal normal vector of the curve. Since there are three families of regular curves in R31 , we briefly recall the Frenet apparatus in each one of the three families [9, 14]
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