About local configurations in arithmetic planes

AbstractIn Vittone and Chassery (Proc. of DGCI’97, Vol. 1347 of Lecture Notes in Computer Sciences, 1997, pp. 87–98) J.-M. Chassery and J. Vittone studied local configurations of (m,n)-cubes in naive planes in function of the parameters of these naive planes. More precisely, they enumerated the bicubes and the tricubes that appear in a naive hyperplane of parameters (a,b,c). A symmetry about the line c=a+b appears clearly in this enumeration. The aim of this paper is to prove that the configurations of (n×n)-cubes in the plane of parameters (a,b,c) are in one-to-one relation with those in the plane of parameters (c-b, c-a,c). If we restrict the parameters to the planes such that a+b⩽c, we note a second symmetry about the line c=2b; We also prove this symmetry. We generalize a theorem established by Réveilles and Gérard (Gérard, Proc. of DGCI’99, Vol. 1568 of Lecture Notes in Computer Sciences, 1999, pp. 65–75, Reveilles, Vision Geometry 4, Vol. 2573 of SPIE 95, San Diego, 1995) and these symmetries to the local configurations of planes of given thickness