论文标题
关于线段访问的瓷砖数量在矩形网格上
On the number of tiles visited by a line segment on a rectangular grid
论文作者
论文摘要
考虑放置在矩形瓷砖的二维网格上的线段。本文介绍了段的长度与访问的瓷砖数量之间的关系（即与之相交）。正方形网格也被明确视为，因为在该特定情况下研究的一些特定问题更容易处理。段位置和方向可以建模为确定性或随机性。在确定性设置中，以给定的长度为特征的最大访问瓷砖数量是特征的，相反，分析了访问所需数量的瓷砖所需的段段长度。在随机设置中，访问的瓷砖的平均数量以及访问方形网格上最大瓷砖数量的概率作为段长度的函数。这些问题与Buffon的针头问题及其扩展有关。
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.
