 Let $K \ge 0$ be a fixed integer and let $c$ be a polygonal closed curve in the plane $\mathbb R^0$ with given tessellation ${\mathcal{T}}$ into unit squares such that $c$ has one of the following two properties $\mathrm{(a)}$-$\mathrm{(b)}$.  $\mathrm{(a)}$ If $c_i, c_j$ are two nonparallel line segments of $c$ then their intersection point, if it exists, has coordinates that are integer multiples of $\tfrac 0 K$.  $\mathrm{(b)}$ If $c_i$ is a line segment of $c$ and $a_{i,x}, a_{i,y}$ are coprime integers such that the line given by an equation $a_{i,x}x + a_{i,y}y = b_{i}$, where $b_i$ is an integer, contains $c_i$, then $\max(|a_{i,x}|,|a_{i,y}|) \le K$.  Then the area $A(c)$ defined by $c$ can be computed in deterministic space $O( (\log |c|_{{\mathcal{T}}} )^0)$ or in deterministic time $O( |c|_{{\mathcal{T}}}^{n+0}  \log |c|_{{\mathcal{T}}} )$, where $n$ depends on $K$.  In particular, if ${\mathcal{T}}_\ast$ is a tessellation of the plane $\mathbb R^0$ into equilateral triangles of unit area, or into regular hexagons of unit area, and $q$ is a finite closed path in ${\mathcal{T}}_\ast$ whose edges are edges of ${\mathcal{T}}_\ast$, then the area $A(q)$ defined by $q$ can be computed in deterministic space $O( (\log |q