Arcs and Curves in Digital Pictures

Characterizations of digital “simple arcs” and “simple closed curves” are given. In particular, it is shown that the following are equivalent for sets <italic>S</italic> having more than four points: (1) <italic>S</italic> is a simple curve; (2) <italic>S</italic> is connected and each point of <italic>S</italic> has exactly two neighbors in <italic>S</italic>; (3) <italic>S</italic> is connected, has exactly one hole, and has no deletable points. It follows that if a “shrinking” algorithm is applied to a connected <italic>S</italic> that has exactly one hole, it shrinks to a simple curve.