[1] W.J. Che and J.C. Paul, Lines of curvature and umbilical points for implicit surfaces,
Comput. Aided Geom. Design, 24 (7), 395–409, 2007.
[2] Ü. Çiftçi, A generalization of Lancret’s theorem, J. Geom. Phys. 59 (12), 1597–1603,
2009.
[3] M.P. do Carmo, Differential Geometry of Curves and Surfaces, Englewood Cliffs:
Prentice Hall, 1976.
[4] N. do Espírito-Santo, S. Fornari, K. Frensel and J. Ripoll, Constant mean curvature
hypersurfaces in a Lie group with a bi-invariant metric, Manuscripta Math. 111 (4),
459–470, 2003.
[5] E. Evren, E. Bayram and E. Kasap, Surface pencil with a common line of curvature
in Minkowski 3-space, Acta Math. Sin. (Engl. Ser.) 30 (12), 2103–2118, 2014.
[6] C.Y. Li, R.H. Wang and C.G. Zhu, Parametric representation of a surface pencil
with a common line of curvature, Comput. Aided Design, 43 (9), 1110–1117, 2011.
[7] C.Y. Li, R.H. Wang and C.G. Zhu, An approach for designing a developable surface
through a given line of curvature, Comput. Aided Design, 45, 621–627, 2013
[8] C.Y. Li, C.G. Zhu and R.H. Wang, A generalization of surface family with common
line of curvature, Appl. Math. Comput. 219 (17), 9500–9507, 2013.
[9] C.Y. Li, C.G. Zhu and R.H. Wang, Spacelike developable surfaces through a common
line of curvature in Minkowski space, J. Adv. Mech. Des. Syst. Manuf. 9 (4), 1–9,
2015.
[10] T. Maekawa, F.E. Wolter and N.M. Patrikalakis, Umbilics and lines of curvature for
shape interrogation, Comput. Aided Geom. Design, 13 (2), 133–161, 1996
[11] T.J. Willmore, An Introduction to Differential Geometry, Oxford University Press,
1959.
[12] D.W. Yoon, General helices of AW(k)-type in the Lie group, J. Appl. Math. 2012,
Art. No. 535123, 2012.
[13] X.P. Zhang, W.J. Che and J.C. Paul, Computing lines of curvature for implicit
surfaces, Comput. Aided Geom. Design, 26 (9), 923–940, 2009.
A generalization for surfaces using a line of curvature in Lie group
In this study, we investigate how to construct surfaces using a line of curvature in a 3-dimensional Lie group. Then, by utilizing the Frenet frame, we give the conditions that a curve becomes a line of curvature on a surface when the marching-scale functions are more general expressions. After then, we provide some crucial examples of how efficient our method is on these surfaces.
[1] W.J. Che and J.C. Paul, Lines of curvature and umbilical points for implicit surfaces,
Comput. Aided Geom. Design, 24 (7), 395–409, 2007.
[2] Ü. Çiftçi, A generalization of Lancret’s theorem, J. Geom. Phys. 59 (12), 1597–1603,
2009.
[3] M.P. do Carmo, Differential Geometry of Curves and Surfaces, Englewood Cliffs:
Prentice Hall, 1976.
[4] N. do Espírito-Santo, S. Fornari, K. Frensel and J. Ripoll, Constant mean curvature
hypersurfaces in a Lie group with a bi-invariant metric, Manuscripta Math. 111 (4),
459–470, 2003.
[5] E. Evren, E. Bayram and E. Kasap, Surface pencil with a common line of curvature
in Minkowski 3-space, Acta Math. Sin. (Engl. Ser.) 30 (12), 2103–2118, 2014.
[6] C.Y. Li, R.H. Wang and C.G. Zhu, Parametric representation of a surface pencil
with a common line of curvature, Comput. Aided Design, 43 (9), 1110–1117, 2011.
[7] C.Y. Li, R.H. Wang and C.G. Zhu, An approach for designing a developable surface
through a given line of curvature, Comput. Aided Design, 45, 621–627, 2013
[8] C.Y. Li, C.G. Zhu and R.H. Wang, A generalization of surface family with common
line of curvature, Appl. Math. Comput. 219 (17), 9500–9507, 2013.
[9] C.Y. Li, C.G. Zhu and R.H. Wang, Spacelike developable surfaces through a common
line of curvature in Minkowski space, J. Adv. Mech. Des. Syst. Manuf. 9 (4), 1–9,
2015.
[10] T. Maekawa, F.E. Wolter and N.M. Patrikalakis, Umbilics and lines of curvature for
shape interrogation, Comput. Aided Geom. Design, 13 (2), 133–161, 1996
[11] T.J. Willmore, An Introduction to Differential Geometry, Oxford University Press,
1959.
[12] D.W. Yoon, General helices of AW(k)-type in the Lie group, J. Appl. Math. 2012,
Art. No. 535123, 2012.
[13] X.P. Zhang, W.J. Che and J.C. Paul, Computing lines of curvature for implicit
surfaces, Comput. Aided Geom. Design, 26 (9), 923–940, 2009.
Yoon, D. W., & Kucukarslan Yuzbasi, Z. (2021). A generalization for surfaces using a line of curvature in Lie group. Hacettepe Journal of Mathematics and Statistics, 50(2), 444-452. https://doi.org/10.15672/hujms.664764
AMA
Yoon DW, Kucukarslan Yuzbasi Z. A generalization for surfaces using a line of curvature in Lie group. Hacettepe Journal of Mathematics and Statistics. April 2021;50(2):444-452. doi:10.15672/hujms.664764
Chicago
Yoon, Dae Won, and Zuhal Kucukarslan Yuzbasi. “A Generalization for Surfaces Using a Line of Curvature in Lie Group”. Hacettepe Journal of Mathematics and Statistics 50, no. 2 (April 2021): 444-52. https://doi.org/10.15672/hujms.664764.
EndNote
Yoon DW, Kucukarslan Yuzbasi Z (April 1, 2021) A generalization for surfaces using a line of curvature in Lie group. Hacettepe Journal of Mathematics and Statistics 50 2 444–452.
IEEE
D. W. Yoon and Z. Kucukarslan Yuzbasi, “A generalization for surfaces using a line of curvature in Lie group”, Hacettepe Journal of Mathematics and Statistics, vol. 50, no. 2, pp. 444–452, 2021, doi: 10.15672/hujms.664764.
ISNAD
Yoon, Dae Won - Kucukarslan Yuzbasi, Zuhal. “A Generalization for Surfaces Using a Line of Curvature in Lie Group”. Hacettepe Journal of Mathematics and Statistics 50/2 (April 2021), 444-452. https://doi.org/10.15672/hujms.664764.
JAMA
Yoon DW, Kucukarslan Yuzbasi Z. A generalization for surfaces using a line of curvature in Lie group. Hacettepe Journal of Mathematics and Statistics. 2021;50:444–452.
MLA
Yoon, Dae Won and Zuhal Kucukarslan Yuzbasi. “A Generalization for Surfaces Using a Line of Curvature in Lie Group”. Hacettepe Journal of Mathematics and Statistics, vol. 50, no. 2, 2021, pp. 444-52, doi:10.15672/hujms.664764.
Vancouver
Yoon DW, Kucukarslan Yuzbasi Z. A generalization for surfaces using a line of curvature in Lie group. Hacettepe Journal of Mathematics and Statistics. 2021;50(2):444-52.