Covering the plane by a sequence of circular disks with a constraint

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 (D-n)(n is an element of N) be a sequence of closed unit circular disks such that Un is an element of N Dn = R-2 with the condition that for n >= 2, the center of the disk D-n lies in Dn-1. What is a "most economical" or an optimal way of placing D-n for all n is an element of 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 D-n, then for all n >= 2,