Characterizations of Unit Darboux Ruled Surface with Quaternions

Year 2023, Issue: 42, 43 - 54, 31.03.2023

Abdussamet Çalışkan

https://doi.org/10.53570/jnt.1194990

Abstract

This paper presents a quaternionic approach to generating and characterizing the ruled surface drawn by the unit Darboux vector. The study derives the Darboux frame of the surface and relates it to the Frenet frame of the base curve. Moreover, it obtains the quaternionic shape operator and its matrix representation using the normal and geodesic curvatures to provide a more detailed analysis. To illustrate the concepts discussed, the paper offers a clear example that will help readers better understand the concepts and showcases the quaternionic shape operator, Gauss curvature, mean curvature, and rotation matrix. Finally, it emphasizes the need for further research on this topic.

Keywords

Gauss curvature, mean curvature, ruled surfaces, quaternions, quaternionic shape operator

References

• A. J. Hanson, Visualing Quaternions, The Morgan Kaufmann Series in Interactive 3D Technology, Elsevier, San Francisco, 2006.
• J. B. Kuipers, Quaternions and Rotation Sequences: A Primer with Applications to Orbits, Aerospace and Virtual Reality, Princeton University Press, New Jersey, 2002.
• K. Shoemake, Animating Rotation with Quaternion Curves, Siggraph Computer Graphics 19 (1985) 245–254.
• K. Bharathi, M. Nagaraj, Quaternion Valued Function of a Real Variable Serret-Frenet Formulae, Indian Journal of Pure and Applied Mathematics 18 (1987) 507–511.
• O. Keçilioğlu, K. İlarslan, Quaternionic Bertrand Curves in Euclidean 4-Space, Bulletin of Mathematical Analysis and Applications 5 (3) (2013) 27–38.
• M. Babaarslan, Y. Yaylı, A New Approach to Constant Slope Surfaces with Quaternions, ISRN Geometry 2012 (2012) Article ID 126358 8 pages.
• S. Şenyurt, A. Çalışkan, The Quaternionic Expression of Ruled Surfaces, Filomat 32 (16) (2018) 403–411.
• A. Çalışkan, S. Şenyurt, The Dual Spatial Quaternionic Expression of Ruled Surfaces, Thermal Science 23 (1) (2019) 403–411.
• A. Çalışkan, Quaternionic and Dual Quaternionic Darboux Ruled Surfaces, Turkish Journal of Mathematics and Computer Science 13 (1) (2021) 106–114.
• Y. Li, Z. Wang, T. Zhao, Geometric Algebra of Singular Ruled Surfaces, Advances in Applied Clifford Algebras 31 (2) (2021) 1–19.
• S. Aslan, Y. Yaylı, Quaternionic Shape Operator, Advances in Applied Clifford Algebras 27 (2017) 2921–2931.
• M. Babaarslan, Y. Yaylı, Split Quaternions and Spacelike Constant Slope Surfaces in Minkowski 3-Space, International Journal of Geometry 2 (1) (2013) 23–33.
• M. Babaarslan, Y. Yaylı, Split Quaternions and Timelike Constant Slope Surfaces in Minkowski 3-Space, International Journal of Geometry 8 (1) (2019) 57–71.
• M. P. Do Carmo, Differential Geometry of Curves and Surfaces, 2nd Edition, Dover Publications, New York, 2016.
• B. S. Ryuh, Robot Trajectory Planning Using the Curvature Theory of Ruled Surfaces, Doctoral Dissertation Purdue University (1989) West Lafayette.
• F. Güler, E. Kasap, A Path Planning Method for Robot End Effector Motion Using the Curvature Theory of the Ruled Surfaces, International Journal of Geometric Methods in Modern Physics 15 (3) (2018) 1850048 15 pages.
• S. Şenyurt, K. Eren, On Ruled Surfaces with a Sannia Frame in Euclidean 3-space, Kyungpook Mathematical Journal 62 (3) (2022) 509–531.
• A. Kelleci, K. Eren, On Evolution of Some Associated Type Ruled Surfaces, Mathematical Sciences and Applications E-Notes 8 (2) (2020) 178–186.
• K. Eren, H. H. Kosal, Evolution of Space Curves and the Special Ruled Surfaces with Modified Orthogonal Frame AIMS Mathematics 5 (3) (2020) 2027–2040.
• M. Bilici, On the Invariants of Ruled Surfaces Generated by the Dual Involute Frenet Trihedron, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 66 (2) (2017) 62–70.
• S. Palavar, M. Bilici, Dual Ruled Surface Constructed by the Pole Curve of the Involute Curve, International Journal of Open Problems in Computer Science and Mathematics 15 (1) (2022) 39–53.
• S. Palavar, M. Bilici, New-Type Tangent Indicatrix of Involute and Ruled Surface According to Blaschke Frame in Dual Space, Maejo International Journal of Science And Technology 16 (3) (2022) 199–207.
• E. Bayram, M. Bilici, Surface Family with a Common Involute Asymptotic Curve, International Journal of Geometric Methods in Modern Physics 13 (5) (2016) 1650062 9 pages.
• M. Bilici, E. Bayram Surface Construction with a Common Involute Line of Curvature, International Journal of Open Problems in Computer Science and Mathematics 14 (2) (2021) 20–31.
• B. O’Neill, Elementary Differential Geometry, 2nd Edition, Elsevier, London, 2006.
• J. Vince, Quaternions for Computer Graphics, Springer-Verlag, London, 2011.