||One of the main challenges in digital geometry is the optimal transcription of
continuous space into digital space. We study the covering problem of Euclidean straight lines and Euclidean hyperplanes using dilations with structuring elements. These structuring elements are given as rectangles and blocks, respectively. The aim is to find an optimal covering of a Euclidean straight line by dilations of two of its discretizations. As a starting point, we will refer to Azriel Rosenfeld’s chord property of 1974. We determine the optimal coverings of a Euclidean straight line using the dilations with structuring element centered at each point of a Euclidean straight line’s discretization, generalizing the results of Jean-Marc Chassery & Isabelle Sivignon 2013. Then we generalize these cases to higher dimensions
and establish a framework where we can get an optimal way to cover a Euclidean hyperplane by a digital one using dilations.