Intrinsic Finite-Step Characterizations of Discrete General Helices
Abstract
We study polygonal general helices in Euclidean three-space using the turning and signed torsion angles of the discrete Frenet frame. For helices whose axis is not orthogonal to the edge tangents, we prove that the global constant-angle condition is equivalent to the existence of a conserved Frenet-frame vector. On the generic branch, elimination of the auxiliary coefficient yields an intrinsic finite-step compatibility relation involving three consecutive turning angles and two consecutive torsion angles. Complementary phase and linear-subspace formulations cover the antipodal-binormal case. These characterizations reconstruct the helical axis and the helix angle and yield a sharp bound for each turning angle. We also give a spherical formulation through the tangent indicatrix: its vertices lie on a plane section of the unit sphere, which is a small circle in the non-orthogonal case, whereas the orthogonal case is exactly planar. A nonconstant Frenet-data example illustrates the criterion. Finally, for uniform chordal sampling of a smooth curve, the discrete Lancret-type quotient and the reconstructed helical direction converge with second-order accuracy.
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BibTeX
@misc{yoon2026intrinsic,
title = {{Intrinsic Finite-Step Characterizations of Discrete General Helices}},
author = {Dae Won Yoon and Chul Woo Lee and Jae won Lee},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00673},
eprint = {2609.00673},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00673},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Intrinsic Finite-Step Characterizations of Discrete General Helices
AU - Dae Won Yoon
AU - Chul Woo Lee
AU - Jae won Lee
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00673
UR - https://arxiv.org/abs/2609.00673
AB - We study polygonal general helices in Euclidean three-space using the turning and signed torsion angles of the discrete Frenet frame. For helices whose axis is not orthogonal to the edge tangents, we prove that the global constant-angle condition is equivalent to the existence of a conserved Frenet-frame vector. On the generic branch, elimination of the auxiliary coefficient yields an intrinsic finite-step compatibility relation involving three consecutive turning angles and two consecutive torsion angles. Complementary phase and linear-subspace formulations cover the antipodal-binormal case. These characterizations reconstruct the helical axis and the helix angle and yield a sharp bound for each turning angle. We also give a spherical formulation through the tangent indicatrix: its vertices lie on a plane section of the unit sphere, which is a small circle in the non-orthogonal case, whereas the orthogonal case is exactly planar. A nonconstant Frenet-data example illustrates the criterion. Finally, for uniform chordal sampling of a smooth curve, the discrete Lancret-type quotient and the reconstructed helical direction converge with second-order accuracy.
ER -