This paper focuses on the on-line recognition of digital arcs. The main contribution is to propose a simple and linear algorithm for three subproblems: on-line recognition of digital arcs coming from the digitization of a disk having (i) a fixed radius, (ii) a boundary that con- tacts a fixed point and (iii) a center that belongs to a fixed straight line. Solving such subproblems is interesting in itself, but also for the recog- nition of digital arcs. Indeed the proposed algorithm can be used as an oracle in multidimensional search techniques or can be iteratively used in a naive manner. Moreover, since the algorithm is on-line, it is a means of segmenting digital curves in a very fast way.