  A turn sequence of left and right turns is realized as a simple rectilinear
chain of integral segments whose turns at its bends are the same as the turn
sequence. The chain starts from the origin and ends at some point which we call
a reachable point of the turn sequence. We investigate the combinatorial and
geometric properties of the set of reachable points of a given turn sequence
such as the shape, connectedness, and sufficient and necessary conditions on
the reachability to the four signed axes. We also prove the upper and lower
bounds on the maximum distance from the origin to the closest reachable point
on signed axes for a turn sequence. The bounds are expressed in terms of the
difference between the number of left and right turns in the sequence as well
as, in certain cases, the length of the maximal monotone prefix or suffix of
the turn sequence. The bounds are exactly matched or tight within additive
constants for some signed axes.
