  Recent advancements in observational techniques have led to new tests of the
general relativistic predictions for black-hole spacetimes in the strong-field
regime. One of the key ingredients for several tests is a metric that allows
for deviations from the Kerr solution but remains free of pathologies outside
its event horizon. Existing metrics that have been used in the literature often
do not satisfy the null convergence condition that is necessary to apply the
strong rigidity theorem and would have allowed us to calculate the location of
the event horizon by identifying it with an appropriate Killing horizon. This
has led earlier calculations of event horizons of parametrically deformed
metrics to either follow numerical techniques or simply search heuristically
for coordinate singularities. We show that several of these metrics, almost by
construction, are circular. We can, therefore, use the weak rigidity and
Carter's rotosurface theorem and calculate algebraically the locations of their
event horizons, without relying on expansions or numerical techniques. We apply
this approach to a number of parametrically deformed metrics, calculate the
locations of their event horizons, and place constraints on the deviation
parameters such that the metrics remain regular outside their horizons. We show
that calculating the angular velocity of the horizon and the effective gravity
there offers new insights into the observational signatures of deformed
metrics, such as the sizes and shapes of the predicted black-hole shadows.
