Gravitational Lensing By a Massive Object in a Dark Matter Halo. II.
  Shear, Phase, and Image Geometry
  We study the gravitational lensing influence of a massive object in a dark
matter halo, using a simple model of a point mass embedded in a spherical
Navarro--Frenk--White halo. Building on the analysis of critical curves and
caustics presented in the first part of this work, we proceed to explore the
geometry of images formed by the lens. First, we analyze several lensing
quantities including shear, phase, and their weak-lensing approximations,
illustrating the results with image-plane maps. We derive formulae and present
a geometric interpretation for the shear and phase of a combination of two
axially symmetric mass distributions. In the case of our lens model, we
describe the occurrence of zero-shear points and specify the conditions under
which they become umbilic points. Second, we use the eigenvalue decomposition
of the inverse of the lens-equation Jacobian matrix to compute the
magnification and flattening of lensed images. Based on this, we introduce the
convergence--shear diagram, a novel and compact way of visualizing the
properties of images formed by a particular gravitational lens. We inspect
relative deviations of the analyzed lensing quantities in order to evaluate the
perturbing effect of the point mass and the applicability of the weak-lensing
approximation. We explore the dependence of the results on the point-mass
parameters by studying grids of plots for different combinations of its
position and mass. We provide analytical explanations for important patterns
arising in these plots and discuss the implications for the lensing influence
of isolated compact bodies in dark matter halos.
