Unified curvature modeling of surface constrained helices and associated ruled surfaces

Emad Solouma; Ghaliah Alhamzi; Mona Bin-Asfour; Sayed Saber; Emad Solouma; Ghaliah Alhamzi; Mona Bin-Asfour; Sayed Saber

doi:10.3934/math.2026324

AIMS Mathematics

2026, Volume 11, Issue 3: 7847-7870. doi: 10.3934/math.2026324

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Research article

Unified curvature modeling of surface constrained helices and associated ruled surfaces

1.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University, Riyadh 11623, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Al-Baha University, Al-Baha 65779, Saudi Arabia

Received: 17 January 2026 Revised: 06 March 2026 Accepted: 17 March 2026 Published: 25 March 2026
MSC : 51M04, 53A04

This paper introduces a new class of surface constrained curves, termed generalized B-helices, in the Euclidean three-space $ \mathrm{E}^3 $. The analysis is developed through a rotational modification of the classical Darboux frame, called the $ \mathbb{B} $-Darboux frame $ \{\mathrm{T}, \chi_1, \chi_2\} $. This frame combines the effects of geodesic and normal curvatures into two coupled invariants, referred to as the $ \mathbb{B} $-Darboux curvatures, providing a unified description of curve geometry on regular surfaces. Within this framework, three types of generalized $ \mathbb{B} $-helices are defined, and their characterization conditions are expressed in terms of these curvature functions. The spherical representations corresponding to the $ \mathbb{B} $-Darboux frame vectors are also examined, and explicit relations for their curvature and torsion are derived. It is shown that each type of generalized $ \mathbb{B} $-helix generates a circular indicatrix on the unit sphere, establishing a direct geometric link between spatial and spherical curves. To validate the theoretical developments, illustrative examples and graphical simulations of ruled surfaces associated with the frame vectors are presented, confirming the applicability and effectiveness of the proposed model.
- $ \mathbb{B} $-Darboux frame,
- generalized $ \mathbb{B} $-helices,
- surface curves,
- Euclidean geometry,
- curvature invariants
Citation: Emad Solouma, Ghaliah Alhamzi, Mona Bin-Asfour, Sayed Saber. Unified curvature modeling of surface constrained helices and associated ruled surfaces[J]. AIMS Mathematics, 2026, 11(3): 7847-7870. doi: 10.3934/math.2026324

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Abstract

This paper introduces a new class of surface constrained curves, termed generalized B-helices, in the Euclidean three-space $ \mathrm{E}^3 $. The analysis is developed through a rotational modification of the classical Darboux frame, called the $ \mathbb{B} $-Darboux frame $ \{\mathrm{T}, \chi_1, \chi_2\} $. This frame combines the effects of geodesic and normal curvatures into two coupled invariants, referred to as the $ \mathbb{B} $-Darboux curvatures, providing a unified description of curve geometry on regular surfaces. Within this framework, three types of generalized $ \mathbb{B} $-helices are defined, and their characterization conditions are expressed in terms of these curvature functions. The spherical representations corresponding to the $ \mathbb{B} $-Darboux frame vectors are also examined, and explicit relations for their curvature and torsion are derived. It is shown that each type of generalized $ \mathbb{B} $-helix generates a circular indicatrix on the unit sphere, establishing a direct geometric link between spatial and spherical curves. To validate the theoretical developments, illustrative examples and graphical simulations of ruled surfaces associated with the frame vectors are presented, confirming the applicability and effectiveness of the proposed model.

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