Interactions between Digital Geometry and Combinatorics on Words

TL;DR

This paper explores the connections between digital geometry and combinatorics on words, highlighting how combinatorial tools can describe geometric features and improve algorithms for digital figures.

Contribution

It introduces a combinatorial approach to digital geometry, linking polyominoes, digital convexity, and tilings with words like Lyndon and Christoffel, and develops efficient algorithms.

Findings

Polyominoes can be encoded by words for geometric analysis.

Lyndon and Christoffel words characterize digital convexity.

Radix-trees enable efficient path intersection detection.

Abstract

We review some recent results in digital geometry obtained by using a combinatorics on words approach to discrete geometry. Motivated on the one hand by the well-known theory of Sturmian words which model conveniently discrete lines in the plane, and on the other hand by the development of digital geometry, this study reveals strong links between the two fields. Discrete figures are identified with polyominoes encoded by words. The combinatorial tools lead to elegant descriptions of geometrical features and efficient algorithms. Among these, radix-trees are useful for efficiently detecting path intersection, Lyndon and Christoffel words appear as the main tools for describing digital convexity; equations on words allow to better understand tilings by translations.