Calculation of the differential equations and harmonicity of the involute curve according to unit Darboux vector with a new method

Abstract

In this study we first write the characterizations of involute of a curve by means of the unit Darboux vector of the involute curve. Then we make use of the Frenet formulas [1] to explain the characterizations of involute of a curve by means of Frenet apparatus of the main curve. Finally we examined the helix as an example.

Keywords

• unit Darboux vector,Laplace operator,biharmonic

Kaynakça

1. Çakır O.,Şenyurt, S. Harmonicity and Differential Equation of Involute of a Curve in E3. Thermal Science. 23(6), 2019, 2119–2125.
2. Boyer, C. A History of Mathematics, New York:Wiley. 1968, 334.
3. Bilici, M., Caliskan, M. On the involutes of the spacelike curve with a timelike binormal in Minkowski 3-space. International Mathematical Forum. 4(31), 2009, 1497–1509.
4. Şenyurt S., Cevahir C., Altun Y. On Spatial Quaternionic Involute Curve A New View. Advances in Clifford Algebras. 27(2), 2017, 1815–1824.
5. Kocayigit, H. and Hacisalihoglu H. H. 1-Type curves and biharmonic curves in Euclidean 3-space. Int. Elect. Journ. of Geo. 4(1), 2011 , 97–101.
6. Arslan, K., Kocayigit, H. and Onder, M. Characterizations of Space Curves with 1-type Darboux Instantaneous Rotation Vector. Commun. Korean Math. Soc. 31 (2), 2016, 379–388.
7. Chen, B. Y. And Ishikawa, S. Biharmonic Surface in Pseudo-Euclidean Spaces. Mem. Fac. Sci. Kyushu Univ. 45(1),1991 , 323–347.
8. Şenyurt, S. , Çakır O. Diferential Equations for a Space Curve According to the Unit Darboux Vector. Turk. J. Math. Comput. Sci. 9(1), 2018, 91–97.

9. Sabuncuoglu, A. Diferensiyel Geometri, Nobel Akademik Yayincilik, Ankara, 2014.
10. Fenchel, W. On The Differential Geometry of Closed Space Curves. Bulletin of the American Mathematical Society. 57, 1951, 44–54.
11. Şenyurt, S., Sivas, S., Çalışkan, A. N∗C∗-Smarandache Curves of Involute-Evolute Curve Couple According to Frenet Frame. Algebras, Groups and Geometries. 33(2), 2016, 153–163.
12. Kocayigit, H. , Önder M., Hacisalihoglu, H.H. Harmonic 1-type Curves andWeak Biharmonic Curves in Lorentzian 3-space. An Alele Stiintifice Ale Universitatii ”Al.I. Cuza” Din Iasi(S.N.) Matematica, Tomul LX. 60(1), 2014, 109–124.