Acquiring periodic tilings of regular polygons from images

Abstract

We describe how we have acquired geometrical models of many periodic tilings of regular polygons from two large collections of images. These models are based on a simplification of the representation recently proposed by us that uses complex numbers. We also describe an algorithm for deciding when two representations give the same tiling, which was used to identify coincidences in these collections.

An integer representation for periodic tilings of the plane by regular polygons

“…A tiling is m-Archimedean when it has m types of vertices. Using our symmetry detection algorithm, we have classified all tilings in the collections acquired [11] and have confirmed the classification given by Galebach [19]. The supplementary material for this paper contains examples of tilings having each of the 17 wallpaper groups.…”

“…Having proven its worth within our previous works [10,11] on rendering and acquisition of tilings, our representation is both convenient and general enough to show great potential for many more operations and analyses of tilings. Accordingly, the remainder of this paper offers for the first time a comprehensive account of the representation, presenting its properties, common topological and geometric operations, and sample applications.…”

“…This paper describes a general, uniform, integerbased representation for periodic tilings of the plane by regular polygons, and discusses its properties in full detail. It significantly expands our previous work on the subject [10,11]. New material includes: full details on the representation and its motivation, an in-depth discussion of its properties, and new algorithms for tiling reconstruction and symmetry detection that are simple and robust, since they do not rely on nearest-neighbors searches and geometric computations with numerical tolerances.…”

“…This completes the description of our representation of tilings. We have acquired [11] and published [9] a large collection of tilings in this representation. We shall now explain how to reconstruct the tiling from a representation.…”

“…Closely related to uniqueness is the issue of deciding whether two representations define the same tiling. This is a crucial task for identifying and removing duplicate tilings when acquiring [11] or generating large collections of tilings by systematic enumeration. The simplest approach to deciding equivalence of two representations is to generate two large vertex clouds, one for each representation, and test whether they match after applying a translation or a rotation.…”