On the Smarandache Curves of Spatial Quaternionic Involute Curve

  • Süleyman ŞenyurtEmail author
  • Ceyda Cevahir
  • Yasin Altun
Review Article


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.


Quaternionic curves Involute curve Quaternionic Smarandache curves 

Mathematics Subject Classification

53A04 53C26 



Authors are also thankful to honorable reviewers for their valuable suggestion which helps to improve the quality of the manuscript.


  1. 1.
    Bharathi K, Nagaraj M (1987) Quaternion valued function of a real variable Serret–Frenet formula. Indian J Pure Appl Math 18(6):507–511MathSciNetzbMATHGoogle Scholar
  2. 2.
    Karadaǧ M, Sivridaǧ Aİ (1997) Characterizations for ouaternionic trendlines. Univ Erciyes J Inst Sci Technol Erciyes Univ 13:23–36Google Scholar
  3. 3.
    Tuna A, Çöken AC (2014) On the quaternionic inclined curves in the semi-Euclidean space \(E_{2}^{4}\). Appl Math Comput 155(2):373–389zbMATHGoogle Scholar
  4. 4.
    Erişir T, Güngör MA (2014) Some characterizations of quaternionic rectifying curves in the semi-Euclidean space \(E_{2}^{4}\). Honam Math J 36(1):67–83MathSciNetCrossRefGoogle Scholar
  5. 5.
    Demir S, Özdaş K (2005) Serret–Frenet formulas with real quaternions. University of Süleyman Demirel J Inst Sci 9(3):1–7zbMATHGoogle Scholar
  6. 6.
    Fenchel W (1951) On the differential geometry of closed space curves. Bull Am Math Soc 57(1):44–54MathSciNetCrossRefGoogle Scholar
  7. 7.
    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):289–294MathSciNetzbMATHGoogle Scholar
  8. 8.
    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):1497–1509MathSciNetzbMATHGoogle Scholar
  9. 9.
    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):2019–2030MathSciNetzbMATHGoogle Scholar
  10. 10.
    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):573–582zbMATHGoogle Scholar
  11. 11.
    Zhang F (1997) Quaternions and matrices of quaternions. Linear Algebra Appl 251:21–57MathSciNetCrossRefGoogle Scholar
  12. 12.
    Soyfidan T (2011) Quaternionic involute-evolute couple curves. Master Thesis, University of SakaryaGoogle Scholar
  13. 13.
    Parlatıcı H (2013) Quaternionic Smarandache curves. Master thesis, University of SakaryaGoogle Scholar
  14. 14.
    Ali AT (2010) Special Smarandache curves in the Euclidean space. Int J Math Comb 2:30–36zbMATHGoogle Scholar
  15. 15.
    Şenyurt S, Sivas S (2013) An application of Smarandache curve (in Turkish). Ordu Univ J Sci Technol 3(1):46–60Google Scholar
  16. 16.
    Şenyurt S, Çalışkan A (2015) An application according to spatial quaternionic Smarandache curve. Appl Math Sci 9(5):219–228Google Scholar
  17. 17.
    Şenyurt S, Grilli L (2015) Spherical indicatrix curves of spatial quaternionic curves. Appl Math Sci 9(90):4469–4477Google Scholar
  18. 18.
    Şenyurt S, Cevahir C, Altun Y (2016) On spatial quaternionic involute curve a new view. Adv Appl Clifford Algebras 27:1815–1824MathSciNetCrossRefGoogle Scholar
  19. 19.
    Hacısalihoǧlu HH (1983) Motion geometry and quaternions theory (in Turkish). University of Gazi Press, AnkaraGoogle Scholar
  20. 20.
    Turgut M, Yılmaz S (2008) Smarandache curves in minkowski spacetime. Int J Math Comb 3:51–55MathSciNetzbMATHGoogle Scholar
  21. 21.
    Çetin M, Kocayiğit H (2013) On the quatenionic Smarandache curves in Euclidean 3-space. Int J Contemp Math Sci 8(3):139–150MathSciNetCrossRefGoogle Scholar

Copyright information

© The National Academy of Sciences, India 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Arts and SciencesOrdu UniverstyOrduTurkey

Personalised recommendations