General helices on a class of generalized cylinders
OPEN MATHEMATICS, cilt.24, sa.1, ss.1-16, 2026 (SCI-Expanded, Scopus)
- Yayın Türü: Makale / Tam Makale
- Cilt numarası: 24 Sayı: 1
- Basım Tarihi: 2026
- Doi Numarası: 10.1515/math-2025-0288
- Dergi Adı: OPEN MATHEMATICS
- Derginin Tarandığı İndeksler: Scopus, Technology Collection (ProQuest), Aerospace Database, Science Citation Index Expanded (SCI-EXPANDED), zbMATH, Directory of Open Access Journals
- Sayfa Sayıları: ss.1-16
- Çanakkale Onsekiz Mart Üniversitesi Adresli: Evet
Özet
We study unit-speed general helices on a class of generalized cylinders in Euclidean 3-space whose base curve is planar and whose ruling direction is orthogonal to the base plane. Instead of relying only on the classical curvature–torsion characterization of general helices, we formulate the helix condition in terms of the surface parametrization and the Darboux frame. This leads to a first-order differential system which gives a local characterization of such helices and provides a constructive method for producing explicit examples. We also obtain a Darboux-data recognition criterion for deciding whether a given unit-speed curve on the cylinder is a general helix. A reduced equation is derived, and its first integral is shown to have a direct geometric interpretation in terms of the component of the helix axis along the ruling direction. As consequences, we obtain restrictions on the first-integral constant and identify the geodesic, asymptotic, and line-of-curvature cases. Several examples, including constant- and nonconstant-curvature base curves, illustrate the theory.