Smarandache Curves for the Integral Curves with the Quasi Frame in Euclidean 3-Space

This study investigates the geometric properties of integral Smarandache curves associated with the quasi-frame in three-dimensional Euclidean space. We derive the Frenet apparatus for four types of integral Smarandache curves: the TNq, TBq, NqBq, and TNqBq Smarandache curves, based on the quasi-frame elements. For each type of curve, we provide explicit formulas for the tangent, normal, and binormal vectors, as well as the curvature and torsion. Furthermore, we establish necessary and sufficient conditions under which these integral Smarandache curves can be classified as general helices or Salkowski curves. Specifically, we show that if the original curve is a general helix, the corresponding integral Smarandache curves also exhibit helical properties. Additionally, we analyze the Darboux vectors associated with these curves and demonstrate their relationship with the Darboux vectors of the original curve, proving that the Darboux vectors coincide when the original curve is a general helix. We also provide illustrative examples, including a circular helix and a space curve, to validate our theoretical findings. These examples demonstrate the application of the quasi-frame in simplifying the analysis of complex curves and highlight the geometric relationships between the original curve and its integral Smarandache curves.