On the number of tiles visited by a line segment on a rectangular grid

TL;DR

This paper investigates how the length and position of a line segment on a rectangular grid affect the number of tiles it intersects, providing both deterministic bounds and probabilistic analysis, especially for square grids.

Contribution

It characterizes maximum tile visits for given segment lengths and analyzes average visits and probabilities in random placements, extending classical geometric probability problems.

Findings

Maximum tiles visited for a given segment length are characterized.

Average number of visited tiles is derived in the random setting.

Probability of visiting the maximum number of tiles is analyzed for square grids.

Abstract

Consider a line segment placed on a two-dimensional grid of rectangular tiles. This paper addresses the relationship between the length of the segment and the number of tiles it visits (i.e. has intersection with). The square grid is also considered explicitly, as some of the specific problems studied are more tractable in that particular case. The segment position and orientation can be modelled as either deterministic or random. In the deterministic setting, the maximum possible number of visited tiles is characterized for a given length, and conversely, the infimum segment length needed to visit a desired number of tiles is analyzed. In the random setting, the average number of visited tiles and the probability of visiting the maximum number of tiles on a square grid are studied as a function of segment length. These questions are related to Buffon's needle problem and its extension by Laplace.