A Note on a Result of Daurat and Nivat

Brlek, Srečko; Labelle, Gilbert; Lacasse, Annie

doi:10.1007/11505877_17

Cited by 9 publications

(11 citation statements)

References 4 publications

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“…2. This result can be extended easily to take into account hexagonal grids in which case the alphabet F must be extended to 6 letters, in which case the Daurat-Nivat relation becomes (see [4,5] for more details)…”

Section: The Daurat-nivat Relationmentioning

confidence: 98%

“…The lower convex hull of S, denoted by Conv − (S), is the clockwise oriented sequence of consecutive edges of Conv(S) starting from the highest vertex and ending on the lowest vertex. [4,5] We recall from Daurat and Nivat [15] that a discrete set E is a subset of Z 2 and an element (i, j) ∈ E corresponds to a unit square with vertices (i ± 1 2 , j ± 1 2 ). One sets P 1/2 = ( 1 2 , 1 2 ) + Z 2 and the salient and reentrant points are defined as follows: Definition 2.…”

Section: Lemmamentioning

confidence: 99%

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Interactions between Digital Geometry and Combinatorics on Words

Brlek

2011

Electron. Proc. Theor. Comput. Sci.

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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

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Section: The Daurat-nivat Relationmentioning

confidence: 98%

Section: Lemmamentioning

confidence: 99%

Interactions between Digital Geometry and Combinatorics on Words

Brlek

2011

Electron. Proc. Theor. Comput. Sci.

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“…A convenient way to represent them is to use the well-known Freeman chain code [1,2] which encodes the contour by a word w on the four letter alphabet Σ = {a, b, a, b}, corresponding to the unit displacements in the four directions (right, up, left, down) on a square grid. Among the many problems that have been considered in the literature, we mention : computations of statistics such as area, moment of inertia [3,4], digital convexity [5,6,7], and tiling of the plane by translation [8,9]. All of the above mentioned problems are solved by using algorithms that are linear in the length of the contour word, but often it is assumed that the path encoded by this word does not intersect itself.…”

Section: Introductionmentioning

confidence: 99%

A Linear Time and Space Algorithm for Detecting Path Intersection

Brlek

Koskas

Provençal

2009

Discrete Geometry for Computer Imagery

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For discrete sets coded by the Freeman chain describing their contour, several linear algorithms have been designed for determining their shape properties. Most of them are based on the assumption that the boundary word forms a closed and non-intersecting discrete curve. In this article, we provide a linear time and space algorithm for deciding whether a path on a square lattice intersects itself. forms the contour of a discrete figure. This is achieved by adding a radix tree structure over a quadtree, where nodes represents grid points, enriched with neighborhood links that are essential for obtaining linearity. Due to its simplicity, this algorithm has many applications and, as an illustrative example, we use it for determining efficiently a solution to the more general problem of multiple paths intersection.

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“…Clearly, the turning number T ( w ) of a closed path w belongs to Z (see [9,10]). In particular, the DauratNivat relation [12] is rephrased as follows.…”

Section: Preliminariesmentioning

confidence: 99%

Combinatorial properties of double square tiles

Massé¹,

Garon²,

Labbé³

2013

Theoretical Computer Science

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We study the combinatorial properties and the problem of generating exhaustively double square tiles, i.e. polyominoes yielding two distinct periodic tilings by translated copies such that every polyomino in the tiling is surrounded by exactly four copies. We show in particular that every prime double square tile may be obtained from the unit square by applying successively some invertible operators on double squares. As a consequence, we prove a conjecture of Provençal and Vuillon [17] stating that these polyominoes are invariant under rotation by angle π.

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A Note on a Result of Daurat and Nivat

Cited by 9 publications

References 4 publications

Interactions between Digital Geometry and Combinatorics on Words

Interactions between Digital Geometry and Combinatorics on Words

A Linear Time and Space Algorithm for Detecting Path Intersection

Combinatorial properties of double square tiles

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