  The analysis of gravitational lensing by galaxies and galaxy clusters
typically relies on ellipsoidal lens models to describe the deflection of light
by the involved dark-matter halos. These models are most often based on the
isothermal density profile -- not an optimal description of the halo, but easy
to use because it leads to an analytic deflection-angle formula. Dark-matter
halos are better described by the Navarro--Frenk--White (NFW) density profile.
We set out to study lensing by a general triaxial ellipsoidal NFW halo, with
the aim of providing an analytic model that would be more consistent with the
current understanding of dark-matter halos. We computed the conversion between
the properties of a triaxial ellipsoidal lens model and its elliptical
surface-density profile. In the case of the NFW lens model, its angular scale
is defined by the projected scale semi-major axis of the halo, while its
lensing regime depends on two parameters: the projected eccentricity $e$ and
the convergence parameter $\kappa_\text{s}$. We employed the Bourassa \&
Kantowski formalism to compute the complex scattering function of the model,
which yields the deflection-angle components when separated into its real and
imaginary parts. We present the obtained closed-form expressions for the
deflection-angle components, valid for an arbitrary eccentricity of the
surface-density profile. We use them to compute and describe the lensing
properties of the model, including: the shear, its components, and the phase;
critical curves, caustics, and the parameter-space mapping of their geometries;
the deformations and orientations of images. The analytically solved
ellipsoidal NFW lens model is available for implementation in gravitational
lensing software. The techniques introduced here such as the image-plane
analysis can prove to be useful for understanding the properties of other lens
models as well.
