Algorithms in combinatorial geometry
Algorithms in combinatorial geometry
Recognizing arithmetic straight lines and planes
DCGA '96 Proceedings of the 6th International Workshop on Discrete Geometry for Computer Imagery
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In classical linear algebra, the question to know if a vector v ∈ Rn belongs to the linear space V ect{v1, v2,...,vk} generated by a familly of vectors, is solved by the Gauss pivot. The problem investigated in this paper is very close to this classical question: we denote ⌊ ċ ⌋ n the function of Rn defined by ⌊ (xi) 1≤i ≤ n ⌋ n = (⌊ xi ⌋) 1 ≤ i ≤ n and the question is now to determine if a given vector v ∈ Zn belongs to ⌊ V ect{v1, v2,..., vk} ⌋ n. This problem can be easily seen as a sytem of inequalities and solved by using linear programming but in some special cases, it can also be seen as a particular geometrical problem and solved by using tools of convex geometry. We will see in this framework that the question v ∈ ⌊ V ect{v1, v2,...., vk} ⌋ n? generalizes the problem of recognition of the finite parts of digital hyperplanes and we will give equivalent formulations which allow to solve it efficiently.