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

Year 2025, Volume: 7 Issue: 1, 15 - 27, 30.06.2025

Melek Erdoğdu

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

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