A new parameterization of digital straight lines

A 1:1 correspondence is established between digital straight lines which start at a fixed point and a simple set of quadruples of integer parameters. Such a representation by parameters is useful for enumeration, First, the authors show a 1:1 correspondence between point pairs in a planar set of points and the linear dichotomies of this set. Then, from the equivalence between digital lines and linear dichotomies of points on the digitization grid, they prove a 1:1 and ´onto´ correspondence between digital straight lines starting at a fixed point and a well-defined set of pairs of grid points. It follows that four parameters uniquely represent any given digital line with a fixed starting point. An O(N) algorithm is given for determining the parameters from the digital line, as well as O(log N) algorithms for transforming between these parameters and the parameters suggested by L. Dorst and A.W.M. Smeulders (1984)