On the Language of Standard Discrete Planes and Surfaces

Abstract

AbstractA standard discrete plane is a subset of ℤ3 verifying the double Diophantine inequality μ ≤ ax + by + cz < μ + ω, with (a,b,c) ≠ (0,0,0). In the present paper we introduce a generalization of this notion, namely the (1,1,1)-discrete surfaces. We first study a combinatorial representation of discrete surfaces as two-dimensional sequences over a three-letter alphabet and show how to use this combinatorial point of view for the recognition problem for these discrete surfaces. We then apply this combinatorial representation to the standard discrete planes and give a first attempt of to generalize the study of the dual space of parameters for the latter [VC00].