The Differential Geometry of a Space Curve via a Constant Vector in ℝ3

The differential geometry of space curves is a fascinating area of research for mathematicians and physicists, and this refers to its crucial applications in many areas. In this paper, a new method is derived to study the differential geometry of space curves. More specifically, the position vector of a constant vector in R3 is given in the Frenet apparatus of a space curve, and it is implemented to study the differential geometry of the given space curve. Easy and neat proofs of various well-known results are given using this new method. Also, new results and the properties of space curves are obtained in light of this new method. More specifically, the position vectors of helices are given in simple forms. Moreover, a new frame associated with a smooth curve is obtained, as well as new curvatures associated with the new frame. The new frame and its curvatures are investigated and used to give the position vector of slant helix in a simple and memorable form. Furthermore, some non-trivial examples are given to illustrate some of the results obtained in this article.