We show that cyclic permutations avoiding $321$ are precisely those permutations whose image under the fundamental bijection avoid a set of vincular patterns. We do this by using pattern functions and arrow patterns, in combination with the characterization of $321$ avoidance in terms of equality of the upper bound of the Daiconis-Graham inequalities. We then explore some consequences of this result, including upper and lower bound results on the growth rate of $321$ avoiding cycles.