Geometric Elements of Constant Precession Curve

Year 2020, Volume: 2 Issue: 2, 48 - 55, 09.12.2020

Emre Öztürk

Abstract

In this paper, we determine the geodesic curvature and the geodesic torsion of the constant precession curve, and the normal curvature of the circular hyperboloid of one-sheet in the direction of tangent vector of the constant precession curve, through the Darboux frame. We give the causal character of the constant precession curve in Minkowski space and we state the constant angle that its principal normal makes with fixed direction. Moreover, we give some angles just as, the angle between the osculating plane of the constant precession curve and the tangent plane to the circular hyperboloid of one-sheet; the angle between principal unit normal of the constant precession curve and unit normal vector of the circular hyperboloid of one-sheet, in terms of curvatures of the curve.

Keywords

Constant precession curve, slant helix, hyperboloid of one sheet

References

• [1] Ali, A. T. (2012). Position vectors of slant helices in Euclidean 3-space. Journal of the Egyptian Mathematical Society, 20, 1-6.
• [2] Ali, A. T., & Turgut, M. (2010). Some characterizations of slant helices in the Euclidean space $\mathbb{E}^{n}$. Hacettepe Journal of Mathematics and Statics, 39(3), 327-336.
• [3] Berger, M., & Gostiaux, B. (1988). Differential Geometry: Manifolds, Curves, and Surfaces. Springer.
• [4] Do Carmo, M. P. (1976). Differential Geometry of Curves and Surfaces. Prentice Hall.
• [5] Izumiya, S., & Takeuchi, N. (2004). New special curves and developable surfaces. Turkish Journal of Mathematics, 28(2), 153-164.
• [6] Kula, L., & Yaylı, Y. (2005). On slant helix and its spherical indicatrix. Applied Mathematics and Computation, 169(1), 600-607.
• [7] O’Neill, B. (1966). Elementary Differential Geometry. USA: Academic Press. New York.
• [8] Oprea, J. (1997). Differential Geometry and its Applications. Prentice-Hall Inc.
• [9] Scofield P. D. (1995). Curves of constant precession. The American Mathematical Monthly, 102(6), 531-537.