Research Article
BibTex RIS Cite

Dual Smarandache curves and ruled surfaces obtained from the alternative frame

Year 2022, Volume: 12 Issue: 2, 512 - 526, 15.04.2022
https://doi.org/10.17714/gumusfenbil.1004096

Abstract

According to the E-Study theorem, the dual Smarandache curve chosen on the dual unit sphere in dual space corresponds to the ruled surface formed by the directional lines in the Euclidean 3-space. In this study, some characterizations of ruled surfaces corresponding to dual Smarandache curves constructed with the help of dual components of the alternative frame are investigated.

References

  • Arslan, İ. (2010). Dual küresel eğriler ve yüzeyler. [Doktora Tezi, Ankara Üniversitesi Fen Bilimleri Enstitüsü].
  • Baky, R.A. (2002). An explicit characterization of dual spherical curve. Communications Faculty of Sciences University of Ankara Series A1-Mathematics and Statistics, 51(2), 1–9. http://dx.doi.org/10.1501/Commua1_0000000356.
  • Bektaş, Ö., & Yüce, S. (2013). Special Smarandache curves according to Darboux frame in E3. Romanian Journal of Mathematics and Computer Science, 3, 48–59.
  • Bishop, R.L. (1975). There is more than one way to frame a curve. The American Mathematical Monthly, 82(3), 246-251. http://dx.doi.org/10.1080/00029890.1975.11993807.
  • Bükcü, B., & Karacan, M.K. (2008). Special Bishop motion and Bishop Darboux rotation axis of the space curve. Journal of Dynamical Systems and Geometric Theories, 6(1), 27-34. http://dx.doi.org/10.1080/1726037X.2008.10698542.
  • Bükcü, B., & Karacan, M.K. (2008). Bishop frame of the spacelike curve with a spacelike principal normal in Minkowski 3-space. Communications Faculty of Sciences University of Ankara Series A1-Mathematics and Statistics, 57(1), 13-22. http://dx.doi.org/10.1501/Commua1_0000000185.
  • Bükcü, B., & Karacan, M.K. (2009). The slant helices according to Bishop frame. International Journal of Mathematics and Computer Science, 3(2), 67-70.
  • Bükcü, B., & Karacan, M.K. (2010). Bishop frame of the spacelike curve with a spacelike binormal in Minkowski 3-space. Selçuk Journal of Applied Mathematics, 11(1), 15-25.
  • Clifford, W. K. (1873). Preliminary sketch of biquaternions. Proceedings of the London Mathematical Society, 4, 361–395.
  • Çalışkan, A., & Şenyurt, S. (2020). Curves and ruled surfaces according to alternative frame in dual space. Communications Faculty of Sciences University of Ankara Series A1-Mathematics and Statistics, 69(1), 684–698. http://dx.doi.org/10.31801/cfsuasmas.487789.
  • Gürses, N.B., Bektaş, O., & Yüce, S., (2016). Special Smarandache curves in R13. Communications Faculty of Sciences University of Ankara Series A1-Mathematics and Statistics, 65(2): 143–160.
  • Hacısalihoğlu, H.H. (Ed.). (1983). Hareket geometrisi ve kuaterniyonlar teorisi. Gazi Üniversitesi Yayınları. Hacısalihoğlu, H.H. (Ed.). (1983). Diferensiyel geometri. İnönü Üniversitesi Yayınları.
  • Izumiya, S., & Takeuchi, N. (2004). New special curves and developable surfaces. Turkish Journal of Mathematics, 28(2), 153-164.
  • Kahraman, T., & Uğurlu, H.H. (2014). Dual Smarandache curves and Smarandache ruled surfaces. Mathematical Sciences and Applications E-Notes, 2(1).
  • Karacan, M.K. (2008). Bishop frame of the timelike curve in Minkowski 3-space. SDÜ Fen Edebiyat Fakültesi Fen Dergisi, 3(1), 80-90.
  • Keskin, O., & Yaylı, Y. (2017). An application of N-Bishop frame to spherical images for direction curves. International Journal of Geometric Methods in Modern Physics, 14(11), 1750162. http://dx.doi.org/10.1142/S0219887817501626.
  • Kuşak Samancı, H., & Kocayiğit, H. (2019). N-Bishop Darboux vector of the spacelike curve with spacelike binormal. Thermal Science, 23(1), 353-360. http://dx.doi.org/10.2298/TSCI181112048K. Scofield, P.D. (1995). Curves of constant precession. The American Mathematical Monthly, 102(6), 531-537. http://dx.doi.org/10.1080/00029890.1995.12004613.
  • Study, E. (Ed.). (1903). Die geometrie der dynamen. Verlag Teubner. Leipzig.
  • Şenyurt, S., & Çalıskan, A. (2015). N*C*- Smarandache curves of Mannheim curve couple according to Frenet frame. International Journal of Mathematical Combinatorics, 1, 1-13. https://doi.org/10.5281/zenodo.815111.
  • Şenyurt, S., Sivas, S., & Çalışkan, A. (2016a). N-C Smarandache curves of involute evolute curve couple according to Frenet frame algebras. Groups And Geometries, 33(2), 153-164.
  • Şenyurt, S., Calıskan, A., & Celik, U. (2016b). N*C*− Smarandache curve of Bertrand curves pair according to Frenet frame. International Journal of Mathematical Combinatorics, 1, 1-7. https://doi.org/10.5281/zenodo.815715.
  • Şenyurt, S., & Kaya, G. (2018). NC-Smarandache curve and NW-Smarandache curve according to alternative frame. Turkish Journal of Mathematics and Computer Science, 10269–274.
  • Şenyurt, S., Çalıskan, A., & Çelik, U. (2021). Smarandache curves of Bertrand curves pair according to Frenet frame. Boletim da Sociedade Paranaense de Matemática, 39 (5), 163-173.
  • Uzunoğlu, B., Gök İ., & Yaylı, Y. (2016). A new approach on curves of constant precession. Applied Mathematics and Computation, 275, 317-323. http://dx.doi.org/10.1016/j.amc.2015.11.083.
  • Yayli, Y., & Saracoğlu, S., (2011). Some notes on dual spherical curves. Journal of Informatics and Mathematical Sciences, 3(2): 177–189. http://dx.doi.org/10.26713%2Fjims.v3i2.49.
  • Yılmaz, S., Özyılmaz, E., & Turgut, M. (2010). New spherical indicatrices and their characterizations. An Saint. University Ovidius Constanta, 18(2), 337-354.
  • Yilmaz, S., & Turgut, M.A. (2010). New version of Bishop frame and an application to spherical images. Journal of Mathematical Analysis and Applications, 371(2),764–776. http://dx.doi.org/10.1016/j.jmaa.2010.06.012.

Alternatif çatıdan elde edilen dual Smarandache eğrileri ve regle yüzeyleri

Year 2022, Volume: 12 Issue: 2, 512 - 526, 15.04.2022
https://doi.org/10.17714/gumusfenbil.1004096

Abstract

E-Study teoremi gereği dual uzayda dual birim küre üzerinde seçilen dual Smarandache eğrisi Öklid-3 uzayındaki yönlü doğruların oluşturmuş olduğu regle yüzeye karşılık gelir. Bu çalışmada alternatif çatının dual bileşenlerinin yardımıyla oluşturulan dual Smarandache eğrilerine karşılık gelen regle yüzeylerine ait bazı karakterizasyonlar incelenmiştir.

References

  • Arslan, İ. (2010). Dual küresel eğriler ve yüzeyler. [Doktora Tezi, Ankara Üniversitesi Fen Bilimleri Enstitüsü].
  • Baky, R.A. (2002). An explicit characterization of dual spherical curve. Communications Faculty of Sciences University of Ankara Series A1-Mathematics and Statistics, 51(2), 1–9. http://dx.doi.org/10.1501/Commua1_0000000356.
  • Bektaş, Ö., & Yüce, S. (2013). Special Smarandache curves according to Darboux frame in E3. Romanian Journal of Mathematics and Computer Science, 3, 48–59.
  • Bishop, R.L. (1975). There is more than one way to frame a curve. The American Mathematical Monthly, 82(3), 246-251. http://dx.doi.org/10.1080/00029890.1975.11993807.
  • Bükcü, B., & Karacan, M.K. (2008). Special Bishop motion and Bishop Darboux rotation axis of the space curve. Journal of Dynamical Systems and Geometric Theories, 6(1), 27-34. http://dx.doi.org/10.1080/1726037X.2008.10698542.
  • Bükcü, B., & Karacan, M.K. (2008). Bishop frame of the spacelike curve with a spacelike principal normal in Minkowski 3-space. Communications Faculty of Sciences University of Ankara Series A1-Mathematics and Statistics, 57(1), 13-22. http://dx.doi.org/10.1501/Commua1_0000000185.
  • Bükcü, B., & Karacan, M.K. (2009). The slant helices according to Bishop frame. International Journal of Mathematics and Computer Science, 3(2), 67-70.
  • Bükcü, B., & Karacan, M.K. (2010). Bishop frame of the spacelike curve with a spacelike binormal in Minkowski 3-space. Selçuk Journal of Applied Mathematics, 11(1), 15-25.
  • Clifford, W. K. (1873). Preliminary sketch of biquaternions. Proceedings of the London Mathematical Society, 4, 361–395.
  • Çalışkan, A., & Şenyurt, S. (2020). Curves and ruled surfaces according to alternative frame in dual space. Communications Faculty of Sciences University of Ankara Series A1-Mathematics and Statistics, 69(1), 684–698. http://dx.doi.org/10.31801/cfsuasmas.487789.
  • Gürses, N.B., Bektaş, O., & Yüce, S., (2016). Special Smarandache curves in R13. Communications Faculty of Sciences University of Ankara Series A1-Mathematics and Statistics, 65(2): 143–160.
  • Hacısalihoğlu, H.H. (Ed.). (1983). Hareket geometrisi ve kuaterniyonlar teorisi. Gazi Üniversitesi Yayınları. Hacısalihoğlu, H.H. (Ed.). (1983). Diferensiyel geometri. İnönü Üniversitesi Yayınları.
  • Izumiya, S., & Takeuchi, N. (2004). New special curves and developable surfaces. Turkish Journal of Mathematics, 28(2), 153-164.
  • Kahraman, T., & Uğurlu, H.H. (2014). Dual Smarandache curves and Smarandache ruled surfaces. Mathematical Sciences and Applications E-Notes, 2(1).
  • Karacan, M.K. (2008). Bishop frame of the timelike curve in Minkowski 3-space. SDÜ Fen Edebiyat Fakültesi Fen Dergisi, 3(1), 80-90.
  • Keskin, O., & Yaylı, Y. (2017). An application of N-Bishop frame to spherical images for direction curves. International Journal of Geometric Methods in Modern Physics, 14(11), 1750162. http://dx.doi.org/10.1142/S0219887817501626.
  • Kuşak Samancı, H., & Kocayiğit, H. (2019). N-Bishop Darboux vector of the spacelike curve with spacelike binormal. Thermal Science, 23(1), 353-360. http://dx.doi.org/10.2298/TSCI181112048K. Scofield, P.D. (1995). Curves of constant precession. The American Mathematical Monthly, 102(6), 531-537. http://dx.doi.org/10.1080/00029890.1995.12004613.
  • Study, E. (Ed.). (1903). Die geometrie der dynamen. Verlag Teubner. Leipzig.
  • Şenyurt, S., & Çalıskan, A. (2015). N*C*- Smarandache curves of Mannheim curve couple according to Frenet frame. International Journal of Mathematical Combinatorics, 1, 1-13. https://doi.org/10.5281/zenodo.815111.
  • Şenyurt, S., Sivas, S., & Çalışkan, A. (2016a). N-C Smarandache curves of involute evolute curve couple according to Frenet frame algebras. Groups And Geometries, 33(2), 153-164.
  • Şenyurt, S., Calıskan, A., & Celik, U. (2016b). N*C*− Smarandache curve of Bertrand curves pair according to Frenet frame. International Journal of Mathematical Combinatorics, 1, 1-7. https://doi.org/10.5281/zenodo.815715.
  • Şenyurt, S., & Kaya, G. (2018). NC-Smarandache curve and NW-Smarandache curve according to alternative frame. Turkish Journal of Mathematics and Computer Science, 10269–274.
  • Şenyurt, S., Çalıskan, A., & Çelik, U. (2021). Smarandache curves of Bertrand curves pair according to Frenet frame. Boletim da Sociedade Paranaense de Matemática, 39 (5), 163-173.
  • Uzunoğlu, B., Gök İ., & Yaylı, Y. (2016). A new approach on curves of constant precession. Applied Mathematics and Computation, 275, 317-323. http://dx.doi.org/10.1016/j.amc.2015.11.083.
  • Yayli, Y., & Saracoğlu, S., (2011). Some notes on dual spherical curves. Journal of Informatics and Mathematical Sciences, 3(2): 177–189. http://dx.doi.org/10.26713%2Fjims.v3i2.49.
  • Yılmaz, S., Özyılmaz, E., & Turgut, M. (2010). New spherical indicatrices and their characterizations. An Saint. University Ovidius Constanta, 18(2), 337-354.
  • Yilmaz, S., & Turgut, M.A. (2010). New version of Bishop frame and an application to spherical images. Journal of Mathematical Analysis and Applications, 371(2),764–776. http://dx.doi.org/10.1016/j.jmaa.2010.06.012.
There are 27 citations in total.

Details

Primary Language Turkish
Journal Section Articles
Authors

Hatice Kusak Samancı 0000-0001-6685-236X

Veysi Cengiz 0000-0001-7843-6793

Publication Date April 15, 2022
Submission Date October 3, 2021
Acceptance Date February 13, 2022
Published in Issue Year 2022 Volume: 12 Issue: 2

Cite

APA Kusak Samancı, H., & Cengiz, V. (2022). Alternatif çatıdan elde edilen dual Smarandache eğrileri ve regle yüzeyleri. Gümüşhane Üniversitesi Fen Bilimleri Dergisi, 12(2), 512-526. https://doi.org/10.17714/gumusfenbil.1004096