Recognizing arithmetic straight lines and planes

Abstract

The problem of recognizing a straight line in the discrete plane ℤ2 (resp. a plane in ℤ3) is to find an algorithm deciding wether a given set of points in ℤ2 (resp. ℤ3) belongs to a line (resp. a plane). In this paper the lines and planes are arithmetic, as defined by Reveilles [Rev91], and the problem is translated, for any width that is a linear function of the coefficients of the normal to the searched line or plane, into the problem of solving a set of linear inequalities. This new problem is solved by using the Fourier's elimination algorithm. If there is a solution, the family of solutions is given by the algorithm as a conjunction of linear inequalities. This method of recognition is well suited to computer imagery, because any traversal algorithm of the given set is possible, and also because any incomplete segment of line or plane can be recognized.