Natural mates of Frenet curves in Euclidean 3-space

For each Frenet curve α in the Euclidean 3-space E³, there exists a unique unit speed curve β tangent to the principal normal vector field of α. We simply call this curve β the natural mate of α. 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.