带约束的圆阵序列覆盖平面
Covering the Plane by a Sequence of Circular Disks with a Constraint
摘要：
我们感兴趣的是下面的问题覆盖的平面上的一系列全等圆盘之间的距离的限制连续磁盘。设$(\mathcal{D}_n)_{n \in \mathbb N}$是一列闭单位圆盘，使得$\cup_{n \in \mathbb{N}} \mathcal{D}_n = \mathbb {R}^2$满足对于$n \ge 2$，圆盘中心$\mathcal{D}_n$位于$\mathcal{D}_{n-1}$中的条件。对于所有$n \in \mathbb{N}$，放置$\mathcal{D}_n$的“最经济”或最佳方法是什么？我们回答这个问题的情况下，没有“急”转是允许的，即如果$C_n$是中心的磁盘$\mathcal{D}_n$，然后为所有$n \ge 2$，% $\angle C_{n-1}C_nC_{n+1}$不是很小。我们还考虑一个相关的问题。我们希望找到一个最优的方法来覆盖平面与单位圆盘的约束，每个磁盘包含至少两个其他磁盘的中心。当圆盘的中心形成二维晶格时，我们找到了答案。
英文摘要：
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 $n \ge 2$, % $\angle C_{n-1}C_nC_{n+1}$ 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
