  The distortion of distances between points under maps is studied. We first
prove a Schwarz-type lemma for quasiregular maps of the unit disk involving the
visual angle metric. Then we investigate conversely the quasiconformality of a
bilipschitz map with respect to the visual angle metric on convex domains. For
the unit ball or half space, we prove that a bilipschitz map with respect to
the visual angle metric is also bilipschitz with respect to the hyperbolic
metric. We also obtain various inequalities relating the visual angle metric to
other metrics such as the distance ratio metric and the quasihyperbolic metric.
