About the maximum cardinality of the digital cover of a curve with a given length

Abstract

We prove that the number of pixels -with pixels as unit lattice squares- of the digitization of a curve (Formula presented.) of Euclidean length l is less than (Formula presented.)+4 which improves by a ratio of (Formula presented.) the previous known bound in 4(Formula presented.) [3]. This new bound is the exact maximum that can be reached. Moreover, we prove that for a given number of squares n, the Minimal Length Covering Curves of n squares are polygonal curves with integer vertices, an appropriate number of diagonal steps and 0, 1 or 2 vertical or horizontal steps. It allows to express the functions N(l), the maximum number of squares that can be crossed by a curve of length l, and L(n), the minimal length necessary to cross n squares. Extensions of these results are discussed with other distances, in higher dimensions and with other digitization schemes.