The aim of this study is to obtain a general version of constant-angle ruled surfaces constructed using a Frenet frame in Galilean space $ \mathbb{G}_{3} $. We define generalized special ruled surfaces by considering cases where the surface normal vectors are parallel to the tangent, principal normal, or binormal vector fields of the base curve. We provide criteria regarding the locus of singular points of these surfaces. Specifically, for a general constant angle ruled surface whose normal vectors are parallel to the binormal vector field $ (\mathcal{M}^{b}) $, we explicitly characterize the singular set as $ \{(s, {-\upsilon(s)}/{\delta_{1}(s)\kappa(s)}):s\in I\} $. We establish analogous singular sets and characterizations for cuspidal edge, swallowtail, and cuspidal butterfly singularities for a surface whose normal vectors are parallel to the tangent vector field $ (\mathcal{M}^{t}) $. Conversely, we conclude that the general constant angle ruled surface whose normal vectors are parallel to the principal normal vector field. $ (\mathcal{M}^{n}) $ has no singular points. Finally, as an application of the findings, we give some illustrated examples of helices with singularities.
Citation: Ümit Tokeşer, Seda Nur Ídrísoğlu. Ruled surfaces as wave fronts in Galilean space $ \mathbb{G}_{3} $ and characterizations of geometric singularities[J]. AIMS Mathematics, 2026, 11(5): 14239-14252. doi: 10.3934/math.2026584
The aim of this study is to obtain a general version of constant-angle ruled surfaces constructed using a Frenet frame in Galilean space $ \mathbb{G}_{3} $. We define generalized special ruled surfaces by considering cases where the surface normal vectors are parallel to the tangent, principal normal, or binormal vector fields of the base curve. We provide criteria regarding the locus of singular points of these surfaces. Specifically, for a general constant angle ruled surface whose normal vectors are parallel to the binormal vector field $ (\mathcal{M}^{b}) $, we explicitly characterize the singular set as $ \{(s, {-\upsilon(s)}/{\delta_{1}(s)\kappa(s)}):s\in I\} $. We establish analogous singular sets and characterizations for cuspidal edge, swallowtail, and cuspidal butterfly singularities for a surface whose normal vectors are parallel to the tangent vector field $ (\mathcal{M}^{t}) $. Conversely, we conclude that the general constant angle ruled surface whose normal vectors are parallel to the principal normal vector field. $ (\mathcal{M}^{n}) $ has no singular points. Finally, as an application of the findings, we give some illustrated examples of helices with singularities.
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Ümit Tokeşer, Seda Nur Ídrísoğlu. Ruled surfaces as wave fronts in Galilean space $ \mathbb{G}_{3} $ and characterizations of geometric singularities[J]. AIMS Mathematics, 2026, 11(5): 14239-14252. doi: 10.3934/math.2026584
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