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Given any $n \in \mathbb{Z}^{+}$, we constructively prove the existence ofcovering paths and circuits in the plane which are characterized by the samelink length of the minimum-link covering trails for the two-dimensional grid$G_n^2 := \{0,1, \dots ,n-1\} \times \{0,1, \dots ,n-1\}$. Furthermore, weintroduce a general algorithm that returns a covering cycle of analogous linklength for any even value of $n$. Finally, we provide the tight upper bound$n^2 - 3 + 5 \cdot \sqrt{2}$ units for the minimum total distance travelled tovisit all the nodes of $G_n^2$ with a minimum-link trail (i.e., a trail with $2\cdot n - 2$ edges if $n$ is above two).

Shortest polygonal chains covering each planar square grid