  Starting with the general description of a moving curve, we have recently
presented a unified formalism to show that three distinct space curve
evolutions can be identified with a given integrable equation. Applying this to
the nonlinear Schr\"{o}dinger equation (NLS), we find three sets of coupled
equations for the evolution of the curvature and the torsion, one set for each
moving curve. The first set is given by the usual Da Rios-Betchov equations.
The velocity at each point of the curve that corresponds to this set is well
known to be a local expression in the curve variables. In contrast, the
velocities of the other two curves are shown to be nonlocal expressions. Each
of the three curves is endowed with a corresponding infinite set of geometric
constraints. These moving space curves are found by using their connection with
the integrable Landau- Lifshitz equation. The three evolving curves
corresponding to the envelope soliton solution of the NLS are presented and
their behaviors compared.
