Erratum to “A further study for the upper bound of the cardinality of Farey vertices and applications in discrete geometry” [J. Algebra Comb. Discrete Appl. 2(3) (2015) 169-190]

The equation (4) on the page 178 of the paper previously published has to be corrected. We had only handled the case of the Farey vertices for which $\min\left(\left\lfloor\dfrac{2m}{sr'}\right\rfloor,\left\lfloor\dfrac{n}{s'r}\right\rfloor \right)\in\mathbb{N}^{*}$. In fact we had to distinguish two cases: \min\left(\left\lfloor\dfrac{2m {sr'}\right\rfloor,\left\lfloor\dfrac{n}{s'r}\right\rfloor \right)\in\mathbb{N}^{*} and $\min\left(\left\lfloor\dfrac{2m}{sr'}\right\rfloor,\left\lfloor\dfrac{n}{s'r}\right\rfloor \right)=0$. However, we highlight the correct results of the original paper and its applications. We underline that in this work, we still brought several contributions. These contributions are: applying the fundamental formulas of Graph Theory to the Farey diagram of order $(m,n)$, finding a good upper bound for the degree of a Farey vertex and the relations between the Farey diagrams and the linear diophantine equations

A further study for the upper bound of the cardinality of Farey vertices and application in discrete geometry

The aim of the paper is to bring new combinatorial analytical properties of the Farey diagrams of order $(m,n)$, which are associated to the $(m,n)$-cubes. The latter are the pieces of discrete planes occurring in discrete geometry, theoretical computer sciences, and combinatorial number theory. We give a new upper bound for the number of Farey vertices $FV(m,n)$ obtained as intersections points of Farey lines (\cite{khoshnoudiradfarey}): \exists C>0, \forall (m,n)\in\mathbb{N}^{*2},\quad \Big|FV(m,n)\Big| \leq C m^2 n^2 (m+n) \ln^2 (mn) Using it, in particular, we show that the number of $(m,n)$-cubes $\mathcal{U}_{m,n}$ verifies: \exists C>0, \forall (m,n)\in\mathbb{N}^{*2},\quad \Big|\mathcal{U}_{m,n}\Big| \leq C m^3 n^3 (m+n) \ln^2 (mn) which is an important improvement of the result previously obtained in ~\cite{daurat_tajine_zouaoui_afpdpare}, which was a polynomial of degree 8. This work uses combinatorics, graph theory, and elementary and analytical number theory

Digital planar surface segmentation using local geometric patterns

International audienceThis paper presents a hybrid two-step method for segmenting a 3D grid-point cloud into planar surfaces by using discrete-geometry results. Digital planes contain a finite number of local geometric patterns (LGPs). Such a LGP possesses a set of normal vectors. By using LGP properties, we first reject non-linear points from a point cloud (edge-based step), and then classify non-rejected points whose LGPs have common normal vectors into a planar-surface-point set (region-based step)