Current theories of arrival time have difficulty explaining performance in the common but neglected case of nonlinear approach. Global tau, a variable supposed to guide time-to-passage (TTP) judgments of objects approaching on linear trajectories, does not apply to circular movement. However, TTP judgments are surprisingly accurate in such cases. We simulated movement through a three-dimensional cloud of point-lights on various circular trajectories. Arrival-time judgments were found to be above chance when observers had to determine which of two expansionless targets would pass them first. Similar to the inside bias observed in heading studies on circular trajectories, observers showed a strong bias to select the target on the inside of their own curved motion path as passing by first. Analysis of the projected target motion revealed that targets on the inside had lower optical velocities and relatively high optical acceleration rates. Empirical TTP judgments agreed best with a strategy based on relative optical velocity changes.