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We extend the notion of canonical ordering (initially developed for planartriangulations and 3-connected planar maps) to cylindric (essentially simple)triangulations and more generally to cylindric (essentially internally)$3$-connected maps. This allows us to extend the incremental straight-linedrawing algorithm of de Fraysseix, Pach and Pollack (in the triangulated case)and of Kant (in the $3$-connected case) to this setting. Precisely, for anycylindric essentially internally $3$-connected map $G$ with $n$ vertices, wecan obtain in linear time a periodic (in $x$) straight-line drawing of $G$ thatis crossing-free and internally (weakly) convex, on a regular grid$\mathbb{Z}/w\mathbb{Z}\times[0..h]$, with $w\leq 2n$ and $h\leq n(2d+1)$,where $d$ is the face-distance between the two boundaries. This also yields anefficient periodic drawing algorithm for graphs on the torus. Precisely, forany essentially $3$-connected map $G$ on the torus (i.e., $3$-connected in theperiodic representation) with $n$ vertices, we can compute in linear time aperiodic straight-line drawing of $G$ that is crossing-free and (weakly)convex, on a periodic regular grid$\mathbb{Z}/w\mathbb{Z}\times\mathbb{Z}/h\mathbb{Z}$, with $w\leq 2n$ and$h\leq 1+2n(c+1)$, where $c$ is the face-width of $G$. Since $c\leq\sqrt{2n}$,the grid area is $O(n^{5/2})$.

Canonical ordering for graphs on the cylinder, with applications to periodic straight-line drawings on the flat cylinder and torus