We are interested in the following problem of covering the plane by a sequence of congruent circular disks with a constraint on the distance between consecutive disks. Let ${({\mathcal{D}}_n)}_{n\in \mathbb{N}}$ be a sequence of closed unit circular disks such that ${\cup }_{n\in \mathbb{N}}{\mathcal{D}}_n={\mathbb{R}}^2$ with the condition that for $n\ge 2$, the center of the disk ${\mathcal{D}}_n$ lies in ${\mathcal{D}}_{n-1}$. What is a “most economical” or an optimal way of placing ${\mathcal{D}}_n$ for all $n\in \mathbb{N}$? We answer this question in the case where no “sharp” turn is allowed, i.e. if $C_n$ is the center of the disk ${\mathcal{D}}_n$, then for all $n\ge 2$, $\angle C_{n-1}{C}_n{C}_{n+1}$ is not very small.

We also consider a related problem. We wish to find out an optimal way to cover the plane with unit circular disks with the constraint that each disk contains the centers of at least two other disks. We find out the answer in the case when the centers of the disks form a two-dimensional lattice.

我们对以下问题感兴趣，这些问题是通过一系列连续圆盘覆盖平面，而连续圆盘之间的距离受到限制。让${（{\mathcal{d}}_{ñ}）}_{ñ\in \mathbb{ñ}}$ 是封闭单元圆盘的序列，使得 ${\cup }_{ñ\in \mathbb{ñ}}{\mathcal{d}}_{ñ}={\mathbb{[R}}^2$ 条件是 $ñ\ge 2$，磁盘的中心 ${\mathcal{d}}_{ñ}$ 在于 ${\mathcal{d}}_{ñ-\mathrm{1个}}$。什么是“最经济”或最佳放置方式${\mathcal{d}}_{ñ}$ 对所有人 $ñ\in \mathbb{ñ}$？在不允许“尖锐”转弯的情况下（例如，如果$C_{ñ}$ 是磁盘的中心 ${\mathcal{d}}_{ñ}$，那么对于所有人 $ñ\ge 2$， $\angle C_{ñ-\mathrm{1个}}{C}_{ñ}{C}_{ñ+\mathrm{1个}}$ 不是很小。

我们还考虑了一个相关的问题。我们希望找到一种用单位圆盘覆盖平面的最佳方法，并限制每个圆盘包含至少两个其他圆盘的中心。当圆盘的中心形成二维晶格时，我们找到了答案。