Applications quasi-affines et pavages du plan discret

Abstract

Nehlig (Applications Affines Discrétes et Antialiassage, Ph.D. Thesis, Université Louis Pasteur, Strasbourg, November 1992. Theoret. Comput. Sci. 156 (1995) 1–38) introduced tilings generated by quasi-affine transformations (QAT). Let us remember that a quasi-affine transformation, or QAT, is a transformation of the discrete plane obtained by composing a rational affine transformation with the usual integer function. So it is defined by [g]:Z2→Z2(x,y)↦x′=ax+by+eωy′=cx+dy+fωwhere a,b,c,d,e,f and ω are integers, ω>0, and [] denotes the usual integer function. The reciprocal image of a point (i,j) by a QAT can contain no, one or several points; this reciprocal image is called paving or tile of index (i,j) and noted Pi,j. The pavings form a periodic tiling of the discrete plane. Nehlig defined the supertile which is a set of tiles containing all generic tiles: so the supertile tiles the discrete plane. He defined also the generic strip which is a part of the supertile sufficient to tile the discrete plane and proved that if ω=ad−bc then the paving of index (0,0) tiles the plane. This study has been done assuming that gcd(a,b)=gcd(b,d)=gcd(c,d)=gcd(c,a)=1 (where gcd denotes the greatest common divisor). The aim of this paper is to go on with this study in the general case. We will determine a set of indices I and two vectors u and v such that each paving of index (i,j) can be obtained by translating a paving of index belonging to I, the translation vector depending on (i,j), u and v. We then call paving-cluster the set of pavings of index belonging to I. When gcd(a,b)=gcd(b,d)=gcd(d,c)=gcd(c,a)=1, the paving-cluster contains the same pavings as the generic strip defined by Nehlig. We also prove that the number of neighbours of a paving varies from four to eight; when ω=ad−bc a paving has 4 or 6 neighbours. This last result is a particular case of Beauquier and Nivat's theorem [1].