Canonical Ordering for Triangulations on the Cylinder, with Applications to Periodic Straight-line Drawings

We extend the notion of canonical orderings to cylindric triangulations. This allows us to extend the incremental straight-line drawing algorithm of de Fraysseix et al. to this setting. Our algorithm yields in linear time a crossing-free straight-line drawing of a cylindric triangulation $T$ with $n$ vertices on a regular grid $Z/wZ\times[0,h]$, with $w\leq 2n$ and $h\leq n(2d+1)$, where $d$ is the (graph-) distance between the two boundaries. As a by-product, we can also obtain in linear time a crossing-free straight-line drawing of a toroidal triangulation with $n$ vertices on a periodic regular grid $Z/wZ\times Z/hZ$, with $w\leq 2n$ and $h\leq 1+n(2c+1)$, where $c$ is the length of a shortest non-contractible cycle. Since $c\leq\sqrt{2n}$, the grid area is $O(n^{5/2})$. Our algorithms apply to any triangulation (whether on the cylinder or on the torus) with no loops nor multiple edges in the periodic representation.