On characterization of curves with Tzitzeica type indicatrices in $E^3$ and $E_1^3$

This study investigated the geometric conditions under which the spherical indicatrices of a regular curve were classified as Tzitzeica curves within the frameworks of both Euclidean 3-space $E^3$ and Minkowski 3-space $E_1^3$. A Tzitzeica curve is defined by the invariant property that the ratio of its torsion to the square of the distance from the origin to its osculating plane remains constant. By utilizing the Frenet-Serret frame, we derived explicit differential characterizations involving the curvature $\kappa$ and torsion $\tau$ for the tangent, principal normal, and binormal indicatrices. Our analysis demonstrates that for general helices in $E^3$, these indicatrices reduced to planar Tzitzeica curves. Furthermore, the study extended these findings to timelike curves in $E_1^3$, where the Lorentzian metric introduces distinct differential behaviors. Finally, we highlighted the analytical complexity of the Tzitzeica condition for spacelike curves in Minkowski space, identifying it as a compelling direction for future research in Lorentzian differential geometry.