Abstract
In this study, the spatial quaternionic curve and the relationship between Frenet frames of involute curve of spatial quaternionic curve are expressed by using the angle between the Darboux vector and binormal vector of the basic curve. Secondly, the Frenet vectors of involute curve are taken as position vector and curvature and torsion of obtained Smarandache curves are calculated. The calculated curvatures and torsions are given depending on Frenet apparatus of basic curve. Finally, an example is given and the shapes of these curves are drawn by using Mapple program.
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References
Bharathi K, Nagaraj M (1987) Quaternion valued function of a real variable Serret–Frenet formula. Indian J Pure Appl Math 18(6):507–511
Karadaǧ M, Sivridaǧ Aİ (1997) Characterizations for ouaternionic trendlines. Univ Erciyes J Inst Sci Technol Erciyes Univ 13:23–36
Tuna A, Çöken AC (2014) On the quaternionic inclined curves in the semi-Euclidean space \(E_{2}^{4}\). Appl Math Comput 155(2):373–389
Erişir T, Güngör MA (2014) Some characterizations of quaternionic rectifying curves in the semi-Euclidean space \(E_{2}^{4}\). Honam Math J 36(1):67–83
Demir S, Özdaş K (2005) Serret–Frenet formulas with real quaternions. University of Süleyman Demirel J Inst Sci 9(3):1–7
Fenchel W (1951) On the differential geometry of closed space curves. Bull Am Math Soc 57(1):44–54
Çalışkan M, Bilici M (2002) Some characterizations for the pair of involute-evolute curves in Euclidean space \(E^{3}\). Bull Pure Appl Sci 21(2):289–294
Bilici M, Çalışkan M (2009) On the involutes of the spacelike curve with a timelike binormal in minkowski 3-space. Int Math Forum 4(31):1497–1509
Bilici M, Çalışkan M (2011) Some new notes on the involutes of the timelike curves in minkowski 3-space. Int J Contemp Math Sci 6(41):2019–2030
Bilici M, Çalışkan M (2018) A new perspective on the involutes of the spacelike curve with a spacelike binormal in minkowski 3-space. J Sci Arts 3(44):573–582
Zhang F (1997) Quaternions and matrices of quaternions. Linear Algebra Appl 251:21–57
Soyfidan T (2011) Quaternionic involute-evolute couple curves. Master Thesis, University of Sakarya
Parlatıcı H (2013) Quaternionic Smarandache curves. Master thesis, University of Sakarya
Ali AT (2010) Special Smarandache curves in the Euclidean space. Int J Math Comb 2:30–36
Şenyurt S, Sivas S (2013) An application of Smarandache curve (in Turkish). Ordu Univ J Sci Technol 3(1):46–60
Şenyurt S, Çalışkan A (2015) An application according to spatial quaternionic Smarandache curve. Appl Math Sci 9(5):219–228
Şenyurt S, Grilli L (2015) Spherical indicatrix curves of spatial quaternionic curves. Appl Math Sci 9(90):4469–4477
Şenyurt S, Cevahir C, Altun Y (2016) On spatial quaternionic involute curve a new view. Adv Appl Clifford Algebras 27:1815–1824
Hacısalihoǧlu HH (1983) Motion geometry and quaternions theory (in Turkish). University of Gazi Press, Ankara
Turgut M, Yılmaz S (2008) Smarandache curves in minkowski spacetime. Int J Math Comb 3:51–55
Çetin M, Kocayiğit H (2013) On the quatenionic Smarandache curves in Euclidean 3-space. Int J Contemp Math Sci 8(3):139–150
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Şenyurt, S., Cevahir, C. & Altun, Y. On the Smarandache Curves of Spatial Quaternionic Involute Curve. Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. 90, 827–837 (2020). https://doi.org/10.1007/s40010-019-00640-5
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DOI: https://doi.org/10.1007/s40010-019-00640-5