Generalized Functionality for Arithmetic Discrete Planes

Valerie Berthe (1) , Christophe Fiorio (1) , Damien Jamet (1)

1 ARITH - Arithmétique informatique

The discrete plane P(a,b,c,mu,omega) is the set of points (x,y,z) in Z^3 satisfying 0 =< ax+by+cz + mu < omega. In the case omega =max (|a|,|b|,|c|), the discrete plane is said naive and is well-known to be functional on a coordinate plane. The aim of our paper is to extend the notion of functionality to a larger family of arithmetic discrete planes by introducing a suitable orthogonal projection direction (alpha,beta,gamma) satisfying alpha a + beta b + gamma c =omega. We then apply this functionality property to the enumeration of some local configurations, that is, the (m,n)-cubes such as introduced in [VitChas99].

Mots clés: Lattice; Local configurations; Digital hyperplane; Digital plane; Digital geometry; Functionality

Domaines: Mathématique discrète [cs.DM]

Valerie Berthe, Christophe Fiorio, Damien Jamet. Generalized Functionality for Arithmetic Discrete Planes. DGCI: Discrete Geometrey for Computer Imagery, Apr 2005, Poitiers, France. pp.276-286, ⟨10.1007/978-3-540-31965-8_26⟩. ⟨hal-00580571⟩