  Vanishing points and vanishing lines are classical geometrical concepts in
perspective cameras that have a lineage dating back to 3 centuries. A vanishing
point is a point on the image plane where parallel lines in 3D space appear to
converge, whereas a vanishing line passes through 2 or more vanishing points.
While such concepts are simple and intuitive in perspective cameras, their
counterparts in catadioptric cameras (obtained using mirrors and lenses) are
more involved. For example, lines in the 3D space map to higher degree curves
in catadioptric cameras. The projection of a set of 3D parallel lines converges
on a single point in perspective images, whereas they converge to more than one
point in catadioptric cameras. To the best of our knowledge, we are not aware
of any systematic development of analytical models for vanishing points and
vanishing curves in different types of catadioptric cameras. In this paper, we
derive parametric equations for vanishing points and vanishing curves using the
calibration parameters, mirror shape coefficients, and direction vectors of
parallel lines in 3D space. We show compelling experimental results on
vanishing point estimation and absolute pose estimation for a wide range of
catadioptric cameras in both simulations and real experiments.
