Consider an arrangement of $k$ lines intersecting the unit square. There is some minimum scaling factor so that any placement of a rectangle with aspect ratio $1 \times p$ with $p\geq 1$ must non-transversely intersect some portion of the arrangement or unit square. Assuming the lines of the arrangement are axis-aligned, we show the optimal arrangement depends on the aspect ratio of the rectangle. In particular, the optimal arrangement is either evenly spaced parallel lines or an evenly spaced grid of lines. We present the precise aspect ratios of rectangles for which each of the two nets is optimal.
翻译:考虑一个将单位方形相交的 $k$ 线条安排。 有一些最小的缩放系数, 以便设置一个矩形, 方差比为 1 美元, 乘以 $p\ geq 1 美元, 方块的某部分不可反向交叉。 假设该安排的线条是轴对齐的, 我们则显示最佳安排取决于矩形的方位比。 特别是, 最佳安排要么是平均间距平行线, 要么是均匀间距的线条格网格。 我们展示了两个网间各为最佳的矩形的精确方位比 。