Research Article
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On Null Cartan Rectifying Isophotic and Rectifying Silhouette Curves Lying on a Timelike Surface in Minkowski Space $\mathbb{E}^3_1$

Year 2024, Volume: 17 Issue: 1, 171 - 183, 23.04.2024

Milica Grbović ćirić Jelena Djordjević Emilija Nesovic

https://doi.org/10.36890/iejg.1447199

Abstract

In this paper, we introduce generalized Darboux frames of the first and the second kind along a null Cartan curve lying on a timelike surface in Minkowski space ${E}^{3}_{1}$ and define null Cartan rectifying isophotic and rectifying silhouette curves in terms of the vector field that belongs to generalized Darboux frame of the first kind. We investigate null Cartan rectifying isophotic and rectifying silhouette curves with constant geodesic curvature $k_g$ and geodesic torsion $\tau_g$ and obtain the parameter equations of their axes. We prove that such curves are the null Cartan helices and the null Cartan cubics. We show that the introduced curves with a non-zero constant curvatures $k_g$ and $\tau_g$ are general helices, relatively normal-slant helices and isophotic curves with respect to the same axis. In particular, we find that null Cartan cubic lying on a timelike surface is rectifying isophotic and rectifying silhouette curve having a spacelike and a lightlike axis. Finally, we give some examples.

Keywords

Generalized Darboux frame null Cartan curve rectifying isophotic curve rectifying silhouette curve

Supporting Institution

Serbian Ministry of Science, Technological Development and Inovations

Project Number

no project number

Thanks

The Authors are very grateful to Editors for giving an opportunity to submit this article for Special Issue dedicated to 80-th birthday of prof. B.Y.Chen.

References

  • [1] Bonnor, W. B.: Null curves in a Minkowski space-time. Tensor. 20 (2), 229–242 (1969).
  • [2] Chen, B. Y.: When does the position vector of a space curve always lie in its rectifying plane? American Mathematical Monthly. 110, 147-152 (2003).
  • [3] Chen, B. Y., Dillen, F.: Rectifying curves as centrodes and extremal curves. Bulletin of Instute of Mathematics Academia Sinica. 33 (2), 77-90 (2005).
  • [4] Chen, B. Y.: Rectifying curves and geodesics on a cone in the Euclidean 3-space. Tamkang Journal of Mathematics. 48 (2), 209–214 (2017).
  • [5] Deshmukh, S., Chen, B. Y., Alshammari S. H.: On rectifying curves in Euclidean 3-space. Turkish Journal of Mathematics. 42, 609–620 (2018).
  • [6] Doğan, F.: Isophote curves on timelike surfaces in Minkowski 3-space, An. ¸Stiin¸t. Univ. Al I Cuza Ia¸si Mat. (N.S.) 63 (1), 133–143 (2017).
  • [7] Doğan, F., Yayli, Y.: On isophote curves and their characterizations. Turkish Journal of Mathematics 39 (5), 650–664 (2015).
  • [8] Djordjevic, J., Nešovi´c, E.: On generalized Darboux frame of a pseudo null curve lying on a lightlike surface in Minkowski 3-space. International Electronic Journal of Geometry. 16 (1), 81–94 (2023).
  • [9] Djordjevic, J., Nešovic, E., Öztürk, U. : On generalized Darboux frame of a spacelike curve lying on a lightlike surface in Minkowski space E31. Turkish Journal of Mathematics. 47, 883–897 (2023).
  • [10] Duggal, K.L., Jin, D.H.: Null Curves and Hypersurfaces of Semi-Riemannian Manifolds. World Scientific Publishing, Singapore (2007).
  • [11] Ferrandez, A., Gimenez, A., Lucas, P.: Journal of Physics A: Mathematical and General Null generalized helices in Lorentz–Minkowski spaces, Journal of Physics. Section A: Math. Gen. 35 (39), 8243 (2002).
  • [12] Grbovic, M., Nešovic, E.: Some relations between rectifying and normal curves in Minkowski 3-space. Mathematical Communications. 17, 655-664 (2012).
  • [13] Hananoi, S., Ito N., Izumiya S.: Spherical Darboux images of curves on surfaces. Beitr Algebra Geom. 56(2), 575–585 (2015).
  • [14] İlarslan, K., Nešovic, E.: Some characterizations of null, pseudo null and partially null rectifying curves in Minkowski space-time. Taiwanese Jornal of Mathematics 12 (5), 1035-1044 (2008).
  • [15] İlarslan, K., Nešovic, E., Petrovic-Torgasev, M.: Some characterizations of rectifying curves in the Minkowski 3-space. Novi Sad Journal of Mathematics. 33 (2), 23–32 (2003).
  • [16] Ilarslan, K., Nešovic, E.: Some characterizations of rectifying curves in the Euclidean space E4. Turkish journal of Mathematics 32 21-30 (2008).
  • [17] Izumiya S., Nabaro AC., Sacramento AJ.: Pseudo-spherical normal Darboux images of curves on a timelike surface in three dimensional Lorentz- Minkowski space. J. Geom Phys. 97 105-118, (2015)
  • [18] Lucas, P., Ortega-Yagües, J.A.: A generalization of the notion of helix. Turkish Journal of Mathematics. 47 (4), 1158–1168 (2023).
  • [19] Lucas, P., Ortega-Yagües, J.A.: Rectifying curves in the three-dimensional sphere. Journal of Mathematical Analysis and Applications. 421, (2), 1855-1868 (2015).
  • [20] Lucas, P., Ortega-Yagües, J.A.: Rectifying Curves in the Three-Dimensional Hyperbolic Space. Mediterranean Journal of Mathematics. 13, 2199–2214 (2016).
  • [21] Macit, N., Düldül, M.: Relatively normal-slant helices lying on a surface and their characterizations. Hacettepe Journal of Mathematics and Statistics. 46 (3), 397–408 (2017).
  • [22] Nešovic, E., Koç Öztürk, E. B., Öztürk, U.: On k–type null Cartan slant helices in Minkowski 3–space. Mathematical Methods in the Applied Sciences. 41 (17), 7583–7598 (2018).
  • [23] Nešovic, E., Öztürk, U., Koç Öztürk, E. B.: Some characterizations of pseudo null isophotic curves in Minkowski 3–space. Journal of Geometry. 112 (2), 13 pages (2021). [24] Nešovi´c, E., Öztürk, U., Koç Öztürk, E. B.: On non-null relatively normal-slant helices in Minkowski 3-space. Filomat. 36 (6), 2051–2062 (2022). [25] O’Neill, B.: Semi-Riemannian geometry with applications to relativity. Academic Press, London (1983). [26] Walrave, J.: Curves and surfaces in Minkowski space. Ph.D. Thesis. Katholieke Universiteit Leuven (1995).

Year 2024, Volume: 17 Issue: 1, 171 - 183, 23.04.2024

Milica Grbović ćirić Jelena Djordjević Emilija Nesovic

https://doi.org/10.36890/iejg.1447199

Abstract

Project Number

no project number

References

  • [1] Bonnor, W. B.: Null curves in a Minkowski space-time. Tensor. 20 (2), 229–242 (1969).
  • [2] Chen, B. Y.: When does the position vector of a space curve always lie in its rectifying plane? American Mathematical Monthly. 110, 147-152 (2003).
  • [3] Chen, B. Y., Dillen, F.: Rectifying curves as centrodes and extremal curves. Bulletin of Instute of Mathematics Academia Sinica. 33 (2), 77-90 (2005).
  • [4] Chen, B. Y.: Rectifying curves and geodesics on a cone in the Euclidean 3-space. Tamkang Journal of Mathematics. 48 (2), 209–214 (2017).
  • [5] Deshmukh, S., Chen, B. Y., Alshammari S. H.: On rectifying curves in Euclidean 3-space. Turkish Journal of Mathematics. 42, 609–620 (2018).
  • [6] Doğan, F.: Isophote curves on timelike surfaces in Minkowski 3-space, An. ¸Stiin¸t. Univ. Al I Cuza Ia¸si Mat. (N.S.) 63 (1), 133–143 (2017).
  • [7] Doğan, F., Yayli, Y.: On isophote curves and their characterizations. Turkish Journal of Mathematics 39 (5), 650–664 (2015).
  • [8] Djordjevic, J., Nešovi´c, E.: On generalized Darboux frame of a pseudo null curve lying on a lightlike surface in Minkowski 3-space. International Electronic Journal of Geometry. 16 (1), 81–94 (2023).
  • [9] Djordjevic, J., Nešovic, E., Öztürk, U. : On generalized Darboux frame of a spacelike curve lying on a lightlike surface in Minkowski space E31. Turkish Journal of Mathematics. 47, 883–897 (2023).
  • [10] Duggal, K.L., Jin, D.H.: Null Curves and Hypersurfaces of Semi-Riemannian Manifolds. World Scientific Publishing, Singapore (2007).
  • [11] Ferrandez, A., Gimenez, A., Lucas, P.: Journal of Physics A: Mathematical and General Null generalized helices in Lorentz–Minkowski spaces, Journal of Physics. Section A: Math. Gen. 35 (39), 8243 (2002).
  • [12] Grbovic, M., Nešovic, E.: Some relations between rectifying and normal curves in Minkowski 3-space. Mathematical Communications. 17, 655-664 (2012).
  • [13] Hananoi, S., Ito N., Izumiya S.: Spherical Darboux images of curves on surfaces. Beitr Algebra Geom. 56(2), 575–585 (2015).
  • [14] İlarslan, K., Nešovic, E.: Some characterizations of null, pseudo null and partially null rectifying curves in Minkowski space-time. Taiwanese Jornal of Mathematics 12 (5), 1035-1044 (2008).
  • [15] İlarslan, K., Nešovic, E., Petrovic-Torgasev, M.: Some characterizations of rectifying curves in the Minkowski 3-space. Novi Sad Journal of Mathematics. 33 (2), 23–32 (2003).
  • [16] Ilarslan, K., Nešovic, E.: Some characterizations of rectifying curves in the Euclidean space E4. Turkish journal of Mathematics 32 21-30 (2008).
  • [17] Izumiya S., Nabaro AC., Sacramento AJ.: Pseudo-spherical normal Darboux images of curves on a timelike surface in three dimensional Lorentz- Minkowski space. J. Geom Phys. 97 105-118, (2015)
  • [18] Lucas, P., Ortega-Yagües, J.A.: A generalization of the notion of helix. Turkish Journal of Mathematics. 47 (4), 1158–1168 (2023).
  • [19] Lucas, P., Ortega-Yagües, J.A.: Rectifying curves in the three-dimensional sphere. Journal of Mathematical Analysis and Applications. 421, (2), 1855-1868 (2015).
  • [20] Lucas, P., Ortega-Yagües, J.A.: Rectifying Curves in the Three-Dimensional Hyperbolic Space. Mediterranean Journal of Mathematics. 13, 2199–2214 (2016).
  • [21] Macit, N., Düldül, M.: Relatively normal-slant helices lying on a surface and their characterizations. Hacettepe Journal of Mathematics and Statistics. 46 (3), 397–408 (2017).
  • [22] Nešovic, E., Koç Öztürk, E. B., Öztürk, U.: On k–type null Cartan slant helices in Minkowski 3–space. Mathematical Methods in the Applied Sciences. 41 (17), 7583–7598 (2018).
  • [23] Nešovic, E., Öztürk, U., Koç Öztürk, E. B.: Some characterizations of pseudo null isophotic curves in Minkowski 3–space. Journal of Geometry. 112 (2), 13 pages (2021). [24] Nešovi´c, E., Öztürk, U., Koç Öztürk, E. B.: On non-null relatively normal-slant helices in Minkowski 3-space. Filomat. 36 (6), 2051–2062 (2022). [25] O’Neill, B.: Semi-Riemannian geometry with applications to relativity. Academic Press, London (1983). [26] Walrave, J.: Curves and surfaces in Minkowski space. Ph.D. Thesis. Katholieke Universiteit Leuven (1995).
There are 23 citations in total.

Details

Primary Language English
Subjects Algebraic and Differential Geometry
Journal Section Research Article
Authors

Milica Grbović ćirić 0000-0001-5166-0082

Jelena Djordjević 0000-0003-3052-6778

Emilija Nesovic 0000-0002-3124-6308

Project Number no project number
Early Pub Date April 6, 2024
Publication Date April 23, 2024
Submission Date March 5, 2024
Acceptance Date March 24, 2024
Published in Issue Year 2024 Volume: 17 Issue: 1

Cite

APA Grbović ćirić, M., Djordjević, J., & Nesovic, E. (2024). On Null Cartan Rectifying Isophotic and Rectifying Silhouette Curves Lying on a Timelike Surface in Minkowski Space $\mathbb{E}^3_1$. International Electronic Journal of Geometry, 17(1), 171-183. https://doi.org/10.36890/iejg.1447199
AMA Grbović ćirić M, Djordjević J, Nesovic E. On Null Cartan Rectifying Isophotic and Rectifying Silhouette Curves Lying on a Timelike Surface in Minkowski Space $\mathbb{E}^3_1$. Int. Electron. J. Geom. April 2024;17(1):171-183. doi:10.36890/iejg.1447199
Chicago Grbović ćirić, Milica, Jelena Djordjević, and Emilija Nesovic. “On Null Cartan Rectifying Isophotic and Rectifying Silhouette Curves Lying on a Timelike Surface in Minkowski Space $\mathbb{E}^3_1$”. International Electronic Journal of Geometry 17, no. 1 (April 2024): 171-83. https://doi.org/10.36890/iejg.1447199.
EndNote Grbović ćirić M, Djordjević J, Nesovic E (April 1, 2024) On Null Cartan Rectifying Isophotic and Rectifying Silhouette Curves Lying on a Timelike Surface in Minkowski Space $\mathbb{E}^3_1$. International Electronic Journal of Geometry 17 1 171–183.
IEEE M. Grbović ćirić, J. Djordjević, and E. Nesovic, “On Null Cartan Rectifying Isophotic and Rectifying Silhouette Curves Lying on a Timelike Surface in Minkowski Space $\mathbb{E}^3_1$”, Int. Electron. J. Geom., vol. 17, no. 1, pp. 171–183, 2024, doi: 10.36890/iejg.1447199.
ISNAD Grbović ćirić, Milica et al. “On Null Cartan Rectifying Isophotic and Rectifying Silhouette Curves Lying on a Timelike Surface in Minkowski Space $\mathbb{E}^3_1$”. International Electronic Journal of Geometry 17/1 (April 2024), 171-183. https://doi.org/10.36890/iejg.1447199.
JAMA Grbović ćirić M, Djordjević J, Nesovic E. On Null Cartan Rectifying Isophotic and Rectifying Silhouette Curves Lying on a Timelike Surface in Minkowski Space $\mathbb{E}^3_1$. Int. Electron. J. Geom. 2024;17:171–183.
MLA Grbović ćirić, Milica et al. “On Null Cartan Rectifying Isophotic and Rectifying Silhouette Curves Lying on a Timelike Surface in Minkowski Space $\mathbb{E}^3_1$”. International Electronic Journal of Geometry, vol. 17, no. 1, 2024, pp. 171-83, doi:10.36890/iejg.1447199.
Vancouver Grbović ćirić M, Djordjević J, Nesovic E. On Null Cartan Rectifying Isophotic and Rectifying Silhouette Curves Lying on a Timelike Surface in Minkowski Space $\mathbb{E}^3_1$. Int. Electron. J. Geom. 2024;17(1):171-83.
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