  We introduce a novel statistical way of analyzing the projected mass
distribution in galaxy lenses based solely on the angular distribution of
images in quads around the lens center. The method requires the knowledge of
the lens center location, but the images' distances from the lens center are
not used at all. If the images of a quad are numbered in order of arrival time,
\theta_1 through \theta_4, and \theta_{ij} is the angle between images i and j,
then we define the 'bisector' plane whose axes are linear combinations of
\theta_{23} and \theta_{14}. The bisector plane of a given lens contains all
the quads produced by the lens. We show empirically that all two-fold symmetric
lenses with convex, i.e. non-wavy or petal-like isodensity contours are
identical in the bisector plane of their quads. We also study lenses with
twisting isodensity contours, lumpy substructure, etc. Our results suggest that
to reproduce the general characteristics of the observed quad population,
kpc-scale substructure must be a common feature of galaxy lenses.

Astrophysics