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A new representation of constant angle surfaces in H²×R with split quaternions

Year 2019, Volume: 68 Issue: 1, 633 - 642, 01.02.2019

Siddika Özkaldı Karakuş

https://doi.org/10.31801/cfsuasmas.456433

Abstract

In this paper we study surfaces in H²×R for which the unit normal makes a constant angle with the R-direction. The main idea is to show that constant angle surfaces in H²×R can be obtained by split quaternion product and the matrix representations. Also we give some related examples with their projections of figures.

Keywords

Constant angle surfaces product manifold split quaternions Minkowski space

References

  • Atalay, G.Ş., Güler, F. and Kasap, E., Spacelike Constant Angle Surfaces in Minkowski 3-space, J. Math. Comput. Sci. 2 (2012), No. 3, 451-461.
  • Aslan, S. and Yaylı, Y., Canal surfaces with quaternions. Adv. Appl. Clifford Algebras (2015).
  • Aslan, S. and Yaylı, Y., Split quaternions and canal surfaces in Minkowski 3-space. Int. J. Geom.5(2), (2016), 51--61.
  • Babaarslan, M. and Yaylı, Y., A new approach to constant slope surfaces with quaternion. ISRN Geom. 2012, Article ID 126358, p. 8 (2012). doi:10.5402/ 2012/126358
  • Babaarslan, M. and Yaylı, Y., Split quaternions and spacelike constant slope surfaces in Minkowski 3-space. Int. J. Geom. Vol. 2, No.1 (2013) 23--33.
  • Babaarslan, M. and Yaylı, Y., Split quaternions and timelike constant slope surfaces in Minkowski 3-space, arXiv:1206.3935 [math-ph]
  • Cermelli, P. and Di Scala A., J., Constant angle surfaces in liquid crystals. Phylos Magazine. 87, (2007) 1871-1888.
  • Dillen, F., Fastenakels, J., Van der Veken J. and Vrancken, L., Constant angle surfaces in S²×R. Monaths. Math. 152, (2007) 89--96.
  • Dillen, F. and Munteanu, M.,I., Constant Angle Surfaces in H²×R. Bull. Braz.Math. Soc. 40, (2009) 85--97.
  • Dillen, F. and Munteanu, M.I., Surfaces in H⁺×R, Pure and Applied Differential Geometry, PADGE 2007, Berichte aus der Mathematik (Shaker Verlag) Eds. Franki Dillen, Ignace Van de Woestyne, (2007) 185-193.
  • Gök, İ., Quaternionic approach of canal surfaces constructed by some new ideas, Adv. Appl. Clifford Algebras, 27 (2017) 1175--1190.
  • Gray, A., Abbena, E. and Salamon, S., Modern Geometry of Curves and Surfaces with Mathematica. 3rd Edition. Studies in Advanced Mathematics. 47, 2006.
  • Güler, F., Şaffak, G. and Kasap, E., Timelike Constant Angle Surfaces in Minkowski Space R₁³, Int. J. Contemp. Math. Sciences, Vol. 6, no. 44, (2011), 2189--2200.
  • Hacısalihoğlu, H.H., Hareket Geometrisi ve Kuaterniyonlar Teorisi. Faculty of Sciences and Arts, University of Gazi Press 1983.
  • Kocakusaklı, E., Tuncer, O.O, Gok, İ. and Yaylı Y., A New Representation of Canal Surfaces with Split Quaternions in Minkowski 3-Space Adv. Appl. Clifford Algebras, 27 (2017) 1387--1409.
  • Kula, L. and Yayli, Y., Split quaternions and rotations in semi Euclidean space E₂⁴. J. Korean Math. Soc. 44(6), (2007) 1313--1327.
  • Lopez, R., and Munteanu, M.I., Constant Angle Surfaces in Minkowski Space, Bulletin of the Belgian Mathematical Society, Volume18, Number 2 (2011), 271-286.
  • Munteanu, M.I. and Nistor, A. I., A New Approach on Constant Angle Surfaces in E³. Turk. J. Math. Vol. 33. (2009) 169-178.
  • Nistor, A., Certain Constant Angle Surfaces Constructed on Curves, Int. Electron. J. Geom., Vol. 4 No.1(2011), 79--87.
  • O'Neill, B., Semi-Riemannian Geometry. Academic Press. New York, 1983.
  • Özdemir, M. and Ergin, A.A., Rotations with unit timelike quaternions in Minkowski 3-space. J. Geom. Phys. 56(2), (2006) 322--326.
  • Özkaldı, S. and Yaylı, Y., Constant Angle Surfaces and Curves in E³, Int. Electron. J. Geom., Vol. 4, No.1 (2011), 70--78.
  • Ratcliffe, J. G., Foundatitons of Hyperbolic Manifolds, Springer-Verlag, New York, 1994.
  • Struik, D.J., Lectures on Classical Differential Geometry. New York. Dover, 1988.
  • Fastenakels, J., Munteanu M. I. and Van der Veken, J., Constant angle surfaces in the Heisenberg group, Acta Math. Sinica - Engl. Series, 27, 4 (2011), 747--756.
  • Onnis I. and Piu P., Constant angle surfaces in the Lorentzian Heisenberg group, Arch. Math. 109 (2017), 575--589.
  • Jia, Yan-Bin, Quaternion and Rotation, Com S 477/577 Notes Sep 5, 2017.
  • Ward, J.P., Quaternions and Cayley Numbers. Kluwer Academic Publishers, Boston 1997.

Year 2019, Volume: 68 Issue: 1, 633 - 642, 01.02.2019

Siddika Özkaldı Karakuş

https://doi.org/10.31801/cfsuasmas.456433

Abstract

References

  • Atalay, G.Ş., Güler, F. and Kasap, E., Spacelike Constant Angle Surfaces in Minkowski 3-space, J. Math. Comput. Sci. 2 (2012), No. 3, 451-461.
  • Aslan, S. and Yaylı, Y., Canal surfaces with quaternions. Adv. Appl. Clifford Algebras (2015).
  • Aslan, S. and Yaylı, Y., Split quaternions and canal surfaces in Minkowski 3-space. Int. J. Geom.5(2), (2016), 51--61.
  • Babaarslan, M. and Yaylı, Y., A new approach to constant slope surfaces with quaternion. ISRN Geom. 2012, Article ID 126358, p. 8 (2012). doi:10.5402/ 2012/126358
  • Babaarslan, M. and Yaylı, Y., Split quaternions and spacelike constant slope surfaces in Minkowski 3-space. Int. J. Geom. Vol. 2, No.1 (2013) 23--33.
  • Babaarslan, M. and Yaylı, Y., Split quaternions and timelike constant slope surfaces in Minkowski 3-space, arXiv:1206.3935 [math-ph]
  • Cermelli, P. and Di Scala A., J., Constant angle surfaces in liquid crystals. Phylos Magazine. 87, (2007) 1871-1888.
  • Dillen, F., Fastenakels, J., Van der Veken J. and Vrancken, L., Constant angle surfaces in S²×R. Monaths. Math. 152, (2007) 89--96.
  • Dillen, F. and Munteanu, M.,I., Constant Angle Surfaces in H²×R. Bull. Braz.Math. Soc. 40, (2009) 85--97.
  • Dillen, F. and Munteanu, M.I., Surfaces in H⁺×R, Pure and Applied Differential Geometry, PADGE 2007, Berichte aus der Mathematik (Shaker Verlag) Eds. Franki Dillen, Ignace Van de Woestyne, (2007) 185-193.
  • Gök, İ., Quaternionic approach of canal surfaces constructed by some new ideas, Adv. Appl. Clifford Algebras, 27 (2017) 1175--1190.
  • Gray, A., Abbena, E. and Salamon, S., Modern Geometry of Curves and Surfaces with Mathematica. 3rd Edition. Studies in Advanced Mathematics. 47, 2006.
  • Güler, F., Şaffak, G. and Kasap, E., Timelike Constant Angle Surfaces in Minkowski Space R₁³, Int. J. Contemp. Math. Sciences, Vol. 6, no. 44, (2011), 2189--2200.
  • Hacısalihoğlu, H.H., Hareket Geometrisi ve Kuaterniyonlar Teorisi. Faculty of Sciences and Arts, University of Gazi Press 1983.
  • Kocakusaklı, E., Tuncer, O.O, Gok, İ. and Yaylı Y., A New Representation of Canal Surfaces with Split Quaternions in Minkowski 3-Space Adv. Appl. Clifford Algebras, 27 (2017) 1387--1409.
  • Kula, L. and Yayli, Y., Split quaternions and rotations in semi Euclidean space E₂⁴. J. Korean Math. Soc. 44(6), (2007) 1313--1327.
  • Lopez, R., and Munteanu, M.I., Constant Angle Surfaces in Minkowski Space, Bulletin of the Belgian Mathematical Society, Volume18, Number 2 (2011), 271-286.
  • Munteanu, M.I. and Nistor, A. I., A New Approach on Constant Angle Surfaces in E³. Turk. J. Math. Vol. 33. (2009) 169-178.
  • Nistor, A., Certain Constant Angle Surfaces Constructed on Curves, Int. Electron. J. Geom., Vol. 4 No.1(2011), 79--87.
  • O'Neill, B., Semi-Riemannian Geometry. Academic Press. New York, 1983.
  • Özdemir, M. and Ergin, A.A., Rotations with unit timelike quaternions in Minkowski 3-space. J. Geom. Phys. 56(2), (2006) 322--326.
  • Özkaldı, S. and Yaylı, Y., Constant Angle Surfaces and Curves in E³, Int. Electron. J. Geom., Vol. 4, No.1 (2011), 70--78.
  • Ratcliffe, J. G., Foundatitons of Hyperbolic Manifolds, Springer-Verlag, New York, 1994.
  • Struik, D.J., Lectures on Classical Differential Geometry. New York. Dover, 1988.
  • Fastenakels, J., Munteanu M. I. and Van der Veken, J., Constant angle surfaces in the Heisenberg group, Acta Math. Sinica - Engl. Series, 27, 4 (2011), 747--756.
  • Onnis I. and Piu P., Constant angle surfaces in the Lorentzian Heisenberg group, Arch. Math. 109 (2017), 575--589.
  • Jia, Yan-Bin, Quaternion and Rotation, Com S 477/577 Notes Sep 5, 2017.
  • Ward, J.P., Quaternions and Cayley Numbers. Kluwer Academic Publishers, Boston 1997.
There are 28 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Review Articles
Authors

Siddika Özkaldı Karakuş 0000-0002-2699-4109

Publication Date February 1, 2019
Submission Date February 9, 2018
Acceptance Date March 8, 2018
Published in Issue Year 2019 Volume: 68 Issue: 1

Cite

APA Özkaldı Karakuş, S. (2019). A new representation of constant angle surfaces in H²×R with split quaternions. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 68(1), 633-642. https://doi.org/10.31801/cfsuasmas.456433
AMA Özkaldı Karakuş S. A new representation of constant angle surfaces in H²×R with split quaternions. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. February 2019;68(1):633-642. doi:10.31801/cfsuasmas.456433
Chicago Özkaldı Karakuş, Siddika. “A New Representation of Constant Angle Surfaces in H²×R With Split Quaternions”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68, no. 1 (February 2019): 633-42. https://doi.org/10.31801/cfsuasmas.456433.
EndNote Özkaldı Karakuş S (February 1, 2019) A new representation of constant angle surfaces in H²×R with split quaternions. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68 1 633–642.
IEEE S. Özkaldı Karakuş, “A new representation of constant angle surfaces in H²×R with split quaternions”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 68, no. 1, pp. 633–642, 2019, doi: 10.31801/cfsuasmas.456433.
ISNAD Özkaldı Karakuş, Siddika. “A New Representation of Constant Angle Surfaces in H²×R With Split Quaternions”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68/1 (February 2019), 633-642. https://doi.org/10.31801/cfsuasmas.456433.
JAMA Özkaldı Karakuş S. A new representation of constant angle surfaces in H²×R with split quaternions. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2019;68:633–642.
MLA Özkaldı Karakuş, Siddika. “A New Representation of Constant Angle Surfaces in H²×R With Split Quaternions”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 68, no. 1, 2019, pp. 633-42, doi:10.31801/cfsuasmas.456433.
Vancouver Özkaldı Karakuş S. A new representation of constant angle surfaces in H²×R with split quaternions. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2019;68(1):633-42.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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