[Paper Review] Smarandache Curves According to Sabban Frame on S^2

Kemal Taşköprü, Murat Tosun|arXiv (Cornell University)|Jun 27, 2012

Mathematics and Applications3 references3 citations

TL;DR

This paper introduces and characterizes three types of Smarandache curves—γt, td, and γtd—on the unit sphere 𝑆² using the Sabban frame, a novel moving frame distinct from the Frenet frame. It derives explicit formulas for the geodesic curvature of these curves in terms of the geodesic curvature 𝜅𝑔 of the original curve, providing a differential geometric framework for studying special spherical curves via frame-based transformations.

ABSTRACT

In this paper, we introduce special Smarandache curves according to Sabban frame on and we give some characterization of Smarandache curves. Besides, we illustrate examples of our results.

Motivation & Objective

• To define and study special Smarandache curves—γt, td, and γtd—on the unit sphere 𝑆² using the Sabban frame instead of the classical Frenet frame.
• To characterize the geometric invariants, particularly the geodesic curvature, of these curves in terms of the original curve’s geodesic curvature 𝜅𝑔.
• To provide explicit differential-geometric formulas for the tangent, normal, and binormal vectors (in the Sabban sense) of the new curves.
• To illustrate the theoretical results with concrete examples, including parametric curves and visualizations on 𝑆².
• To extend the theory of Smarandache curves to spherical geometry using a frame that naturally respects the sphere’s intrinsic geometry.

Proposed method

• Define the Sabban frame {γ, t, d} on 𝑆², where t is the unit tangent and d = γ × t, with geodesic curvature 𝜅𝑔 = ⟨t′, d⟩.
• Construct γt-Smarandache curves as β(s*) = (1/√2)(γ + t), and similarly define td and γtd curves via normalized linear combinations of frame vectors.
• Compute the arc-length parameter ratio ds*/ds using the norm of the derivative of β, leading to expressions involving 𝜅𝑔 and its derivative 𝜅𝑔′.
• Derive the tangent vector t𝛽 of β using the chain rule and normalization, expressing it as a linear combination of γ, t, and d.
• Differentiate t𝛽 with respect to s to compute t𝛽′, and express the result in terms of γ, t, d with coefficients involving 𝜅𝑔 and 𝜅𝑔′.
• Compute the geodesic curvature 𝜅𝑔𝛽 = ⟨t𝛽′, d𝛽⟩, where d𝛽 = β × t𝛽, yielding closed-form expressions in terms of 𝜅𝑔 and 𝜅𝑔′.

Experimental results

Research questions

• RQ1 How can Smarandache curves be defined on the unit sphere 𝑆² using the Sabban frame instead of the Frenet frame?
• RQ2 What are the explicit differential-geometric invariants—particularly the geodesic curvature—of γt, td, and γtd-Smarandache curves on 𝑆²?
• RQ3 How does the geodesic curvature of the new curves depend on the geodesic curvature 𝜅𝑔 of the original curve?
• RQ4 Can the transformation from the original curve to its Smarandache counterparts be systematically parameterized and analyzed using the Sabban frame?
• RQ5 What are the geometric and topological implications of constructing curves as linear combinations of position, tangent, and normal vectors in the Sabban frame on 𝑆²?

Key findings

• The γt-Smarandache curve is defined as β(s*) = (1/√2)(γ + t), and its geodesic curvature is given by 𝜅𝑔𝛽 = (1/(2 + 𝜅𝑔²)^{3/2}) × [𝜆₁𝜅𝑔 + 𝜆₂(−1 − 𝜅𝑔) + 2𝜆₃], where 𝜆₁, 𝜆₂, 𝜆₃ are expressions in 𝜅𝑔 and 𝜅𝑔′.
• For the td-Smarandache curve β(s*) = (1/√2)(t + d), the geodesic curvature is 𝜅𝑔𝛽 = (1/(1 + 2𝜅𝑔²)^{3/2}) × [𝜆₁𝜅𝑔 − 𝜆₂ + 𝜆₃(1 + 𝜅𝑔)], with 𝜆₁, 𝜆₂, 𝜆₃ depending on 𝜅𝑔 and 𝜅𝑔′.
• The γtd-Smarandache curve β(s*) = (1/√3)(γ + t + d) has geodesic curvature 𝜅𝑔𝛽 = (1/(4√2(1 − 𝜅𝑔 + 𝜅𝑔²)^{3/2})) × [𝜆₁(2𝜅𝑔 − 1) + 𝜆₂(−1 − 𝜅𝑔) + 𝜆₃(2 − 𝜅𝑔)], with coefficients involving 𝜅𝑔 and 𝜅𝑔′.
• The arc-length parameter ratio is derived as ds*/ds = √[(2 + 𝜅𝑔²)/2] for γt curves, √[(1 + 2𝜅𝑔²)/2] for td curves, and √[2(1 − 𝜅𝑔 + 𝜅𝑔²)/3] for γtd curves.
• The tangent vector of the β curve is expressed as t𝛽 = (1/√(2 + 𝜅𝑔²)) × (−γ + t + 𝜅𝑔 d) for γt curves, and similar normalized linear combinations for other types.
• An explicit example is provided: γ(s) = (cos(s)tanh(s), sin(s)tanh(s), sech(s)), which is used to generate and visualize γt, td, and γtd-Smarandache curves on 𝑆².