Direction Curves of Vectorial Moments with Respect to an Alternative Frame in Euclidean Space

This study introduces an alternative geometric framework for analyzing curves in three-dimensional Euclidean space. By constructing a set of alternative frame vectors and examining their corresponding derivative relations, a new perspective on the differential geometry of space curves is developed. Utilizing these frame vectors and a specifically defined Darboux vector, the vectorial moment curves associated with the original curve are derived. A key contribution of this work is the identification and characterization of the dual directed curves corresponding to these vectorial moments within the alternative frame structure. This approach offers new insights into the intrinsic properties of space curves and broadens the analytical tools available for applications in theoretical and applied geometry.