Abstract
For each Frenet curve a in the Euclidean 3-space E-3, there exists a unique unit speed curve beta tangent to the principal normal vector field of alpha. We simply call this curve beta the natural mate of alpha. The main purpose of this paper is to prove some relationships between a Frenet curve and its natural mate. In particular, we obtain some necessary and sufficient conditions for the natural mate of a Frenet curve to be a helix, a spherical curve, or a curve of constant curvature. Several applications of our main results are also presented.