Space curves in Hahn calculus: rotation structure, a fundamental theorem, and q-deformed invariants at the fixed point
Abstract
We develop the differential geometry of space curves in Hahn's quantum calculus, where the derivative is the difference quotient along the contraction σ(t)=qt+ω with fixed point ω0=ω/(1−q). The Hahn Frenet frame is transported along every orbit of σ by rotations through turning and torsion angles, which identifies it with the discrete Frenet frame of the inscribed orbit polygon. An embedded regular Hahn curve is determined, up to linear reparametrization about ω0, by its trace and the induced contraction. We prove a sharp planarity criterion and a fundamental theorem via an infinite product of rotations and the Jackson--Nörlund integral. For smooth curves, expansions uniform in q show that at ω0 the Hahn curvature and torsion carry the factors [2]q/2 and [3]q/3. These factors, a sampling effect of geometrically decaying steps, disappear under edge-length normalization, but their quotient persists in the limiting ratio of torsion and turning angles of every polygon inscribed in a smooth curve with monotonically converging vertices and edge lengths decaying with ratio~q. In uniformly sampled relaxation towards a stable node, the factors can become measurable biases. We characterize generalized Hahn helices by a two-step relation between turning and torsion angles; their Lancret ratio at ω0 is ±([2]q/2)cotϑ, where ϑ is the helix angle. Unlike the classical case, a constant ratio of torsion and curvature does not characterize helices: curves with constant curvature and torsion are never generalized Hahn helices, and, near ω0, unit-speed helices with a given ratio are unique up to similarities and linear reparametrization.
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BibTeX
@misc{selvi2026space,
title = {{Space curves in Hahn calculus: rotation structure, a fundamental theorem, and $q$-deformed invariants at the fixed point}},
author = {Umut Selvi and Derya Sağlam},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33084},
eprint = {2610.33084},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33084},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Space curves in Hahn calculus: rotation structure, a fundamental theorem, and q-deformed invariants at the fixed point
AU - Umut Selvi
AU - Derya Sağlam
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.33084
UR - https://arxiv.org/abs/2610.33084
AB - We develop the differential geometry of space curves in Hahn's quantum calculus, where the derivative is the difference quotient along the contraction σ(t)=qt+ω with fixed point ω_0=ω/(1−q). The Hahn Frenet frame is transported along every orbit of σ by rotations through turning and torsion angles, which identifies it with the discrete Frenet frame of the inscribed orbit polygon. An embedded regular Hahn curve is determined, up to linear reparametrization about ω_0, by its trace and the induced contraction. We prove a sharp planarity criterion and a fundamental theorem via an infinite product of rotations and the Jackson--Nörlund integral. For smooth curves, expansions uniform in q show that at ω_0 the Hahn curvature and torsion carry the factors [2]_q/2 and [3]_q/3. These factors, a sampling effect of geometrically decaying steps, disappear under edge-length normalization, but their quotient persists in the limiting ratio of torsion and turning angles of every polygon inscribed in a smooth curve with monotonically converging vertices and edge lengths decaying with ratio~q. In uniformly sampled relaxation towards a stable node, the factors can become measurable biases. We characterize generalized Hahn helices by a two-step relation between turning and torsion angles; their Lancret ratio at ω_0 is ±([2]_q/2)cotϑ, where ϑ is the helix angle. Unlike the classical case, a constant ratio of torsion and curvature does not characterize helices: curves with constant curvature and torsion are never generalized Hahn helices, and, near ω_0, unit-speed helices with a given ratio are unique up to similarities and linear reparametrization.
ER -