Construction and Analysis of Smarandache Curves for Integral Binormal Curves in Euclidean 3-Space

This paper presents a detailed geometric analysis of Smarandache curves generated from integral binormal curves within three-dimensional Euclidean space. We provide a complete derivation of the Frenet apparatus encompassing tangent, normal, and binormal vectors, alongside curvature and torsion functions for four distinct types of Smarandache curves: $TN$, $TB$, $NB$, and $TNB$. Furthermore, we establish the necessary and sufficient criteria for these curves to be characterized as general helices or Salkowski curves. A significant outcome of our work is the demonstration that helical characteristics are transmitted from the original curve to its Smarandache derivatives. The theoretical framework is substantiated with numerical examples, including a circular helix and other spatial curves.