Applications quasi affines: pavages par images réciproques

Abstract

Using floating point operations in computer graphics produces uncontrollable rounding errors that can induce dramatic mistakes, especially when topological decisions depend on numeric results. These numerical problems have been reported early [30, 31]. It is thus compulsory to take into account the discrete nature of computers in order to produce a robust computer science theory giving algorithms free of chaotic numerical perturbations. We describe some aspects of a discrete geometry, developed in our research centre [3, 10, 11, 19, 23, 24] based on the arithmetic properties of the integers and free of rounding error problems. The aim of this communication is to study tilings of the discrete plane by reciprocal images of a quasi-affine transformation (QAT) defined on Z 2 by x′ = [ ax + by + e ω ] , y′ = [ cx + dy + f ω ] , where all terms are integers, ω > 0, and the brackets denote usual integer part function. Studying these tilings is possible because of a fundamental one-to-one relationship between these reciprocal images and intersections of discrete lines [27]. The discrete line D of slope − a b , with lower bound m and thickness ω, is the set of integer points ( x, y) such that: m ⩽ ax + by < m + ω, where: a, b, m, ω are integers, ω is positive a and b are relative primes [27] (this is a generalization of Bresenham's discrete lines [7]). The intersection of two nonparallel general discrete lines defines a discrete parallelogram that can either be an empty set of points, or contain one point, or even contain several points. Moreover, these points, if any, are not necessarily connected. First, the dynamical system associated to a QAT reveals rich combinatorial structures that can be somewhat harnessed by regularity theorems. The dynamical system associated to a QAT F is defined by iterations of the points of Z 2: ( X n ) n ⩾0 , with X 0 ϵ N 2, X n > 0 = F( X n−1 ). This dynamical system induces a partitioning of Z 2 into attraction basins. Attraction basins can be split into trees attached on limit cycles. A great variety of situations occurs here at any level: the number of attraction basins as well as the number of points of limit cycles can be any strictly positive number. There can be zero or more trees attached to a limit cycle. We show that the leaves of these trees (i.e. the entry points of the dynamical system) can be seen as the union of n subgroups of Z 2 and define thus a regular pattern over Z 2. The main results in the second part of this paper concern tilings by reciprocal images. Let F be the QAT: x′ = [ ax + by ω ] , y′ = [ cx + dy ω ] . let P i, j = F −1( i, j), the reciprocal image of ( i, j) by F. P i, j is a tile in the tiling of Z 2. We prove that if ω = ¦ad−bc¦, then the tile P 0,0, is a tiling pattern and can be replicated to tile Z 2. When there is no other constraint on ω than being strictly positive, there exists a supertile containing all the generic tiles of the reciprocal image tiling. This supertile is the set of all the points ( x, y) such as S( F( x, y)) = ( i, j), for any given( i, j) ϵ Z 2, with S(i,j) = ([ (di − bj) δ ] , [ (− ci + aj) δ] ) , and δ = ¦ad − bc¦. A supertile contains ω 2 points and no more that δ tiles. A second interesting set of reciprocal image tiles is the generic strip containing following tiles: { P 0,i¦i ϵ [0, δ gcd(ω,δ) ]} . This set contains δ gcd (ω,δ) distinct reciprocal images (with gcd denoting the greatest common divisor) and ω 2 gcd ((ω,δ) points. Both the supertile and the generic strip are tools that aid in understanding the structure of tilings of the discrete plane by reciprocal images of QATs.