Metric properties, i.e. properties which are invariant relative to the group E(3) of all Euclidean motions of the 3-dimensional Euclidean space of fuzzy curves are investigated in this paper. E(3)-equivalence of fuzzy curves is introduced. E(3)-equivalence of fuzzy curves is given in terms of fuzzy curvature and fuzzy torsion. All correlations between these fuzzy invariants of a fuzzy curve are described.