  The iterated composition of two operators, both of which are involutions and
translation invariant, partitions the set of lattice points in the plane into
an infinite sequence of discrete parabolas. Each such parabola contains an
associated stairway-like path connecting certain points on it, induced by the
alternating application of the aforementioned operators. Any two lattice points
in the plane can be connected by paths along the square grid composed of steps
either on these stairways or towards taxicab neighbors. This leads to the
notion of the parabolic-taxicab distance between two lattice points, obtained
as the minimum number of steps of this kind needed to reach one point from the
other.
  In this paper, we describe patterns generated by points on paths of bounded
parabolic-taxicab length and provide a complete description of the balls
centered at the origin. In particular, we prove an earlier conjecture on the
area of these balls.
