  We prove the generalized induction equation and the generalized local
induction equation (GLIE), which replaces the commonly used local induction
approximation (LIA) to simulate the dynamics of vortex lines and thus
superfluid turbulence. We show that the LIA is, without in fact any
approximation at all, a general feature of the velocity field induced by any
length of a curved vortex filament. Specifically, the LIA states that the
velocity field induced by a curved vortex filament is asymmetric in the
binormal direction. Up to a potential term, the induced incompressible field is
given by the Biot-Savart integral, where we recall that there is a direct
analogy between hydrodynamics and magnetostatics. Series approximations to the
Biot-Savart integrand indicate a logarithmic divergence of the local field in
the binormal direction. While this is qualitatively correct, LIA lacks metrics
quantifying its small parameters. Regardless, LIA is used in vortex filament
methods simulating the self-induced motion of quantized vortices. With numerics
in mind, we represent the binormal field in terms of incomplete elliptic
integrals, which is valid for $\mathbb{R}^{3}$. From this and known expansions
we derive the GLIE, asymptotic for local field points. Like the LIA,
generalized induction shows a persistent binormal deviation in the local-field
but unlike the LIA, the GLIE provides bounds on the truncated remainder. As an
application, we adapt formulae from vortex filament methods to the GLIE for
future use in these methods. Other examples we consider include vortex rings,
relevant for both superfluid $^4$He and Bose-Einstein condensates.
