  This paper presents an algorithm to generate a new kind of polygonal mesh
obtained from triangulations. Each polygon is built from a terminal-edge region
surrounded by edges that are not the longest-edge of any of the two triangles
that share them. The algorithm is termed Polylla and is divided into three
phases. The first phase consists of labeling each edge of the input
triangulation according to its size; the second phase builds polygons (simple
or not) from terminal-edges regions using the label system; and the third phase
transforms each non simple polygon into simple ones. The final mesh contains
polygons with convex and non convex shape. Since Voronoi based meshes are
currently the most used polygonal meshes, we compare some geometric properties
of our meshes against constrained Voronoi meshes. Several experiments were run
to compare the shape and size of polygons, the number of final mesh points and
polygons. For the same input, Polylla meshes contain less polygons than Voronoi
meshes, and the algorithm is simpler and faster than the algorithm to generate
constrained Voronoi meshes. Finally, we have validated Polylla meshes by
solving the Laplace equation on an L-shaped domain using the Virtual Element
Method (VEM). We show that the numerical performance of the VEM using Polylla
meshes and Voronoi meshes is similar.
