SOME CHARACTERIZATIONS OF EULER SPIRALS IN E3

Abstract

In this study, some characterizations of Euler spirals in E3 1 have been presented by using their main property that their curvatures are linear. Moreover, discussing some properties of Bertrand curves and helices, the relationship between these special curves in E3 1 have been investigated with di erent theorems and examples. The approach we used in this paper is useful in understanding the role of Euler spirals in E3 1 in di erential geometry.

Keywords

• Curvature,Cornu spiral,Bertrand curve pair

References

1. [1] Harary, G., Tal, A., 3D Euler Spirals for 3D Curve Completion, Symposium on Computational Geometry 2010: 107-108.
2. [2] Harary, G., Tal, A., The Natural 3D Spiral, Computer Graphics Forum, Volume 30(2011), Number 2: 237-246.
3. [3] Kalkan, B., Lopez, R., Spacelike surfaces in Minkowski space satisfying a linear relation between their principal curvatures, Dierential Geometry-Dynamical Systems, Vol.13, 2011, pp. 107-116.
4. [4] K. Ilarslan, E. Nesovic, and M. Petrovic-Torgasev, Some Characterizations of Rectifying Curves in the Minkowski 3-space, Novi Sad J. Math. 33 (2003), no. 2, 23f32g.
5. [5] Lopez, R., Dierential Geometry of Curves and Surfaces in Lorentz-Minkowski Space. [arXiv:0810.3351v1] math.DG, 2008.
6. [6] Saracoglu Celik, S., Yayli, Y., Guler, E., On Generalized Euler Spirals in E3, International Journal of Geometry (Accepted).
7. [7] http://www.cs.iastate.edu/~cs577/handouts/curvature.pdf.