Recognizing arithmetic straight lines and planes

Abstract. The discrete plane P (a, b, c, µ, ω) is the set of points (x, y, z) ∈ Z 3 satisfying 0 ≤ ax+by+cz+µ < ω. In the case ω = 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 (α, β, γ) satisfying αa + βb + γc = ω. We then apply this functionality property to the enumeration of some local configurations, that is, the (m, n)-cubes such as introduced in [VC99].Keywords: digital planes; arithmetic planes; local configurations; functionality of discrete planes.The discrete plane P (a, b, c, µ, ω) is the set of integer points (x, y, z) ∈ Z 3 satisfying 0 ≤ ax + by + cz + µ < ω. In the case ω = max (|a|, |b|, |c|), the discrete plane is said naive and is well-known to be functional on one of the coordinate planes, that is, for any point of P of this coordinate plane, there exists a unique point in the discrete plane obtained by adding to P a third coordinate. Naive planes have been widely studied, see for instance [Rev91, DRR94, DR95, AAS97, VC97, Col02, BB02].The present paper extends the notion of functionality for naive discrete planes to a larger family of arithmetic discrete planes. For that purpose, instead of projecting on a coordinate space, we introduce a suitable orthogonal projection on a plane along a direction (α, β, γ), in some sense dual to the normal vector of the discrete plane P (a, b, c, µ, ω), that is, αa + βb + γc = ω, so that the projection of Z 3 and the points of the discrete plane are in one-to-one correspondence. One interest of the notion of functionality is that it reduces a threedimensional problem to a two-dimensional one, allowing a better understanding of the combinatorial and geometric properties of discrete planes. We thus apply this functionality property to the enumeration of some local configurations, the (m, n)-cubes, for a large family of arithmetic discrete planes, following the approach of [Vui99,BV01].For clarity issues, we have chosen to work here in a three-dimensional space but all the results and methods presented extend in a natural way to R n , with n ≥ 2, as well as to arithmetic discrete lines.

Section: Discussionmentioning

confidence: 99%

Generalized Functionality for Arithmetic Discrete Planes

Berthé

Fiorio

Jamet

2005

“…Such tools are for example the Fourier-Motzkin [10] system simplification algorithm, the Simplex algorithm or the Megiddo's algorithm [17]. Note that the complexity of the Megiddo's algorithm is linear in the number of inequations but the problem comes with the dimension of the system.…”

Section: Visibility Algorithmmentioning

confidence: 99%

“…Many algorithms exist for the DSL recognition problem. Some of these approaches are based on chain code analysis [24], on links between the chain code and arithmetical properties of DSL [6,7], on links between the chain code and the feasible region in the dual -or parameter-space [9,15,23] and others on linear programming tools such that Fourier-Motzkin's algorithm [10]. All these algorithms present a solution either to decide if a given set of pixels is a discrete straight segment (DSS for short) or to segment a discrete curve into DSS, or both.…”

Section: Introductionmentioning

confidence: 99%

Visibility in Discrete Geometry: An Application to Discrete Geodesic Paths

Cœurjolly

2002

“…A number of algorithms exploit the idea to reduce the problem to a relevant linear program and solve it by employing existing methods from linear programming. [10] suggests a method by converting DPS to a system of m 2 linear inequalities, where m is the cardinality of the given set of points. The system is solved by the Fourier elimination algorithm.…”

Section: Introductionmentioning

confidence: 99%

Complexity Analysis for Digital Hyperplane Recognition in Arbitrary Fixed Dimension

Brimkov

Dantchev

2005

Abstract. We consider the following problem. Given a set of pointswhether M is a portion of a digital hyperplane and, if so, determine its analytical representation. In our setting p 1 , p 2 , . . . , p m may be arbitrary points (possibly, with rational and/or irrational coefficients) and the dimension n may be any arbitrary fixed integer. We provide an algorithm that solves this digital hyperplane recognition problem by reducing it to an integer linear programming problem of fixed dimension within an algebraic model of computation. The algorithm performs O (m log D) arithmetic operations, where D is a bound on the norm of the domain elements.