Non-null Space Curves Related by a Transformation of Combescure

Abstract

This study investigates non-null space curves in Minkowski 3-space that are related through the Combescure transformation, a geometric operation characterized by parallel tangent vectors at corresponding points. Using the Frenet apparatus, conditions for two non-null curves to exhibit this transformation are derived, with a focus on their tangent, principal normal, and binormal vectors. The relationships reveal significant properties of curve interactions, including parallelism of Frenet vectors and connections to Bertrand curve pairs. Additionally, the effects of the Combescure transformation on curvature and torsion are analyzed and illustrated through examples. Furthermore, as an application, the conditions under which a spatial curve associated with a non-null biharmonic curve via the Combescure transformation remains biharmonic are determined, supported by relevant examples.