A STUDY on TB-SMARANDACHE TUBE SURFACES in THREE-DIMENSIONAL EUCLIDEAN SPACE

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Yüksel N., Oğraş N.

NEW CONCEPTS AND ADV NATURAL SCIENCE AND MATHEMATICS ,, Prof. Dr. Canan ÖZDEMİR, Editör, ALL SCIENCES ACADEMY, Konya, ss.237-253, 2025

• Yayın Türü: Kitapta Bölüm / Diğer
• Basım Tarihi: 2025
• Yayınevi: ALL SCIENCES ACADEMY
• Basıldığı Şehir: Konya
• Sayfa Sayıları: ss.237-253
• Editörler: Prof. Dr. Canan ÖZDEMİR, Editör
• Erciyes Üniversitesi Adresli: Evet

Özet

In this study, a new type of tube surface is constructed by taking the  TB-Smarandache curve as the center curve in Euclidean 3-space. The geometric behavior of this surface is analyzed within the framework of the Frenet–Serret apparatus. The first and second fundamental forms of the TB- Smarandache tube surface are explicitly calculated, and the corresponding Gaussian and mean curvatures are derived. Based on these results, the conditions under which the surface becomes flat or minimal are examined in detail. Furthermore, the asymptotic and geodesic curvatures of the surface are investigated, and the nature of its curvature lines is discussed. Special attention is given to the characterization of the surface under Weingarten and linear Weingarten conditions. These classifications provide insight into the intrinsic and extrinsic geometry of the surface and help identify special cases with simplified curvature behavior.

An illustrative example is presented to demonstrate the theoretical results, and the corresponding surface is visualized graphically. The findings contribute to the broader understanding of Smarandache-type constructions in differential geometry and may offer potential applications in geometric modeling, computer-aided design, and the study of space curves with frame-dependent properties..