What question did this study set out to answer?

The aim is to explore the relationships between non-null space curves through the Combescure transformation and its impact on their properties.

April 6, 2026Open Access

Non-null space curves related by a transformation of combescure

Key Points

The aim is to explore the relationships between non-null space curves through the Combescure transformation and its impact on their properties.
Derived conditions for non-null curves to exhibit the Combescure transformation using the Frenet apparatus.
Analyzed the relationships of tangent, normal, and binormal vectors.
Explored curvature and torsion effects resulting from the transformation.
Identified significant properties of curve interactions, including the parallelism of Frenet vectors.
Established connections to Bertrand curve pairs.
Determined conditions for retaining biharmonic characteristics in a spatial curve following the transformation.

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 correspond-ing 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 re-veal 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 and supported by relevant examples.

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