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Abstract: In this paper, we study the relationship between Euclidean and discrete space. We study discrete operations based on Euclidean functions: discrete smooth scaling and discrete-continuous rotation. Conversely, we study Euclidean operations based on discrete functions: the discrete based simplification, the Euclidean-discrete union and the Euclidean-discrete co-refinement. These operations operate partly in discrete, and partly in continuous space. Especially for the discrete smooth scaling operation, we provide error bounds when such different operations are chained. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to show that the supercover of an -flat (i.e. a Euclidean affine subspace of dimension ) in Euclidean -space is a discrete analytical object. The supercover of a Euclidean object is a discrete object consisting of all the voxels that intersect . A discrete analytical object is a set of discrete points that is defined by a finite set of inequalities. A method to determine the inequalities defining the supercover of an -flat is provided. [Copyright &y& Elsevier]

Abstract: In this article, we describe the theoretical foundations of the -arithmetization. This method provides a multi-scale discretization of a continuous function that is a solution of a differential equation. This discretization process is based on the Harthong–Reeb line . The Harthong–Reeb line is a linear space that is both discrete and continuous. This strange line stems from a nonstandard point of view on arithmetic based, in this paper, on the concept of -numbers introduced by Laugwitz and Schmieden. After a full description of this nonstandard background and of the first properties of , we introduce the -arithmetization and we apply it to some significant examples. An important point is that the constructive properties of our approach leads to algorithms which can be exactly translated into functional computer programs without uncontrolled numerical error. Afterwards, we investigate to what extent fits Bridges’s axioms of the constructive continuum. Finally, we give an overview of a formalization of the Harthong–Reeb line with the Coq proof assistant. [Copyright &y& Elsevier]