Discrete-Euclidean operations

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