Analytical Formulation of Non-Lightlike Offset Curves and Their Bertrand Structure in Minkowski Space

Abstract

This paper investigates the geometric structure of offset curves derived from non-lightlike curves within the framework of Minkowski three-dimensional space. Offset curves, which consist of points located at a fixed distance along the normal direction from a base curve, are extended to Lorentzian geometry by analyzing their behavior under three different causal character cases: timelike, spacelike with timelike principal normal, and spacelike with timelike binormal. For each case, the curvature, torsion, and arc length of the offset curve are expressed in terms of the corresponding quantities of the original curve and two constants satisfying a specific linear condition that characterizes Bertrand curves. It is shown that the constructed offset curves preserve the principal normal direction of the base curve and satisfy the Bertrand condition, thus forming Bertrand pairs. The results provide a generalization of classical offset theory from Euclidean to Lorentzian geometry and offer new insights into the differential geometry of curves in pseudo-Riemannian spaces, with potential applications in relativistic kinematics and theoretical physics.

Keywords

• Bertrand Curve
• Offset Curve
• Minkowski Space
• Lorentzian Geometry
• Non-lightlike Curve

References

1. Pham, B. (1992). Offset curves and surfaces: a brief survey, Computer-Aided Design. 24 (4), 223–239.
2. Maekawa, T. (1999). An overview of offset curves and surfaces. Computer-Aided Design. 31 (3), 165–173.
3. Arrondo, E., Sendra, J., & Sendrab, R. (1999). Genus formula for generalized offset curves. Journal of Pure and Applied Algebra, 136 (3), 199–209.
4. Newton, W. F. (1898). Theoretical and practical graphics. Macmillan. ISBN 1-113-74312-3.
5. Devadoss, S.L., & Joseph O’Rourke. (2011). Discrete and computational geometry. Princeton University Press. 128–129.
6. Sendra, J. R., Winkler, F., & Perez-Diaz, S. (2008). Rational algebraic curves: a computer algebra approach. Springer Science & Business Media.
7. Schief, W. K. (2003). On the integrability of Bertrand curves and Razzaboni surfaces. Journal of Geometry and Physics, 45(1-2), 130–150.
8. Abdel-Baky, R. A., & Mofarreh, F. (2022). A study on the Bertrand offsets of timelike ruled surfaces in Minkowski 3-space. Symmetry, 14(4).

9. Bilgin, B., & Camcı, Ç. (2022). Timelike V-Bertrand curves in Minkowski 3-Space E^3_1 . Journal of New Theory, 38, 14–24.
10. Babaarslan, M., & Yaylı, Y. (2017). On space-like constant slope surfaces and Bertrand curves in Minkowski 3-space. Analele Stiintifice Ale Universitatii Al I Cuza Din Iasi Matematica, 63(2), 323—339.
11. Inoguchi, J. (1998). Timelike surfaces of constant mean curvature in Minkowski 3-space. Tokyo Journal of Mathematics, 21 (1), 141–152.
12. Özdemir, M., & Ergin, A. A. (2006). Rotations with unit timelike quaternions in Minkowski 3-space. Journal of Geometry and Physics, 56 (2), 322–336.
13. Özdemir, M., Erdoğdu, M., & Şimşek, H. (2014). On the eigenvalues and eigenvectors of a Lorentzian rotation matrix by using split quaternions. Advances in Applied Clifford Algebras, 24, 179–192.
14. Özdemir, M., & Ergin, A. A. (2008). Parallel frames of non-lightlike curves. Missouri Journal of Mathematical Sciences, 20(2), 127–137.
15. Inoguchi, J. (2003). Biharmonic curves in Minkowski 3-space. International Journal of Matehmatics and Mathematical Sciences, 21, 1365–1368.
16. Erdoğdu, M., & Özdemir, M. (2014). Geometry of Hasimoto surfaces in Minkowski 3-space. Mathematical Physics, Analysis and Geometry, 17, 169–181.
17. Ding, Q., & Inoguchi, J. (2004). Schr ¨odinger flows, binormal motion for curves and second AKNS-hierarchies. Chaos Solitons & Fractals, 21 (3), 669–677.