  The local induction equation, or the binormal flow on space curves is a
well-known model of deformation of space curves as it describes the dynamics of
vortex filaments, and the complex curvature is governed by the nonlinear
Schr\"odinger equation. In this paper, we present its discrete analogue,
namely, a model of deformation of discrete space curves by the discrete
nonlinear Schr\"odinger equation. We also present explicit formulas for both
smooth and discrete curves in terms of $\tau$ functions of the two-component KP
hierarchy.
