Reachability of turn sequences

TL;DR

This paper studies the geometric and combinatorial properties of the set of points reachable by rectilinear chains following a given turn sequence, providing bounds on the maximum distance to the origin and conditions for reachability.

Contribution

It introduces new bounds and conditions for the reachability of points on signed axes for turn sequences, advancing understanding of their geometric structure.

Findings

Derived upper and lower bounds on maximum distance from origin to reachable points.

Established necessary and sufficient conditions for reachability on signed axes.

Analyzed shape, connectedness, and geometric properties of reachable point sets.

Abstract

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.