Torsion of a Unit Curve with Constant Angle to its Binormal

International Journal of Mathematics Trends and Technology (IJMTT)
© 2022 by IJMTT Journal
Volume-68 Issue-6
Year of Publication : 2022
Authors : Paul Ryan A. Longhas, Alsafat M. Abdul

How to Cite?

Paul Ryan A. Longhas, Alsafat M. Abdul, " Torsion of a Unit Curve with Constant Angle to its Binormal ," International Journal of Mathematics Trends and Technology, vol. 68, no. 6, pp. 169-172, 2022. Crossref,

In this article, we characterized all unit regular curve with constant angle to its binormal in terms of its torsion. Furthermore, we proved that if α is a unit regular curve defined on open interval with constant angle to its Binormal vector, then its torsion is equal to 0. Consequently, we give a condition when the tangent vector T of α lie in a plane.

Keywords : Torsion, Curve, Binormal, Constant Angle, Curvature.


[1] Nguyen T.P, Debled-Rennesson I, “Curvature and Torsion Estimators for 3D Curves, In et al. Advances in Visual Computing, ISVC, Lecture Notes in Computer Science, Springer, Berlin, Heidelberg, vol. 5358, 2008.
[2] Thomas Lewiner, João D. Gomes, Hélio Lopes, Marcos Craizer, “Curvature, and Torsion Estimators Based on Parametric Curve Fitting,” Computers & Graphics, vol. 29, no. 5, pp. 641-655, 2005.
[3] Manuel Gonzalez-Espinoza, Giovanni Otalora, LucilaKraiselburd, and Susana Landau, “Parametrized Post-Newtonian Formalism in Higher-Order Teleparallel Gravity,” Journal of Cosmology and Astroparticle Physics, 2022.
[4] Ogievetskij V.I & Sokachev Eh. S, “Torsion and Curvature in Terms of the Axial Superfield,” Nuclear physic, vol. 32, no. 3, pp. 870- 879, 1980.
[5] Dragon N, “Torsion and Curvature in Extended Supergravity,” Z. Phys. C - Particles and Fields, vol. 2, pp. 29–32, 1979.
[6] Peter Baekler and Friedrich W Hehl, “Beyond Einstein–Cartan Gravity: Quadratic Torsion and Curvature Invariants with Even and Odd Parity Including All Boundary Terms,” Class. Quantum Grav. vol. 28, pp. 215017, 2011.
[7] Rainer W. Kuhne, “Gauge Theory of Gravity Requires Massive Torsion Field,” International Journal of Modern Physics A, vol. 14, no. 16, pp. 2531-2535, 1999.
[8] Gammack D & Hydon P, “Flow in Pipes with Non-Uniform Curvature and Torsion,” Journal of Fluid Mechanics, vol. 433, pp. 357- 382, 2001.
[9] Bolinder C, “First- and Higher-Order Effects of Curvature and Torsion on the Flow in a Helical Rectangular Duct,” Journal of Fluid Mechanics, vol. 314, pp. 113-138, 1996.
[10] Betchov R, On the Curvature and Torsion of an Isolated Vortex Filament,” Journal of Fluid Mechanics, vol. 22, no. 3, pp. 471-479, 1965.
[11] J. Maeck and G. De Roeck, “Dynamic Bending and Torsion Stiffness Derivation from Modal Curvature and Torsion Rates,” Journal of Sound and Vibration, vol. 225, no. 1, pp. 153-170, 1999.
[12] Saridakis E, Myrzakul S, Myrzakulov K and Yerzhanov K, “Cosmological Applications of f(r,t) Gravity with Dynamical Curvature and Torsion,” American Physical Society, vol. 102, no. 2, pp. 023525, 2000.
[13] He L, Tan H & Huang ZC, Online Handwritten Signature Verification Based on Association of Curvature and Torsion Feature with Hausdorff Distance,” Multimed Tools Appl, vol. 78, pp. 19253–19278, 2019.
[14] Fang L, Lu W, Huang W, “Estimate Algorithms and Embedded Crafts of Curvature and Torsion,” Journal of Graphics, vol. 33, no. 2, pp. 9–13, 2012.
[15] Pişcoran LI, Mishra V.N,”S-Curvature for a New Class of (Α,Β)-Metrics,” RACSAM, vol. 111, pp. 1187–1200, 2017.
[16] Spivak, Michael, “A Comprehensive Introduction to Differential Geometry,” Publish or Perish, Inc, vol. 2, 1999.
[17] Sternberg, Shlomo, “Lectures on Differential Geometry,” Prentice-Hall, 1964.
[18] Struik, Dirk J, “Lectures on Classical Differential Geometry,” Reading, Mass: Addison-Wesley, 1961.
[19] Kühnel, Wolfgang, “Differential Geometry, Student Mathematical Library," Providence, R.I.: American Mathematical Society, vol. 16, pp. 1882174, 2002.
[20] Hanson A.J, "Quaternion Frenet Frames: Making Optimal Tubes and Ribbons from Curves (PDF),” Indiana University Technical Report, 2007.
[21] Frenet F, “On Double Curvature Curves (PDF), Thesis, Toulouse,” Abstract in Journal of Pure and Applied Mathematics, vol. 17, 1852.
[22] Guggenheimer, Heinrich, “Differential Geometry,” Dover, 1977.
[23] Serret J. A, "On Some Formulas Relating to the Theory of Double Curvature Curves (PDF),” Journal of Pure and Applied Mathematics, vol. 16, 1851.
[24] Jordan, Camille, "On the Theory of Curves in N-Dimensional Space", C. R. Acad. Sci. Paris, vol. 79, pp. 795–797, 1874.
[25] Etgen Garret, Hille Einar, Salas Saturnino, Salas and Hille's “Calculus — One and Several Variables 7th ed.,” John Wiley & Sons, pp. 896, 1995.