Digital lines with irrational slopes

How to construct a digitization of a straight line and be able to recognize a straight line in a set of pixels are very important topics in computer graphics. The aim of the present paper is to give a mathematically exact and consistent description of digital straight lines according to Rosenfeld’s definition. The digitizations of the lines with slopes 0 < a < 1, where a is irrational, are considered. We formulate a definition of digitization runs, formulate and prove theorems containing necessary and sufficient conditions for digital straightness. The proof was successfully constructed using only methods of elementary mathematics. The developed and proved theory can be used in the research into the theory of digital lines, their symmetries, translations etc.