If participants used the tau-equalization strategy to steer toward the target, they would steer to equalize the time to closure of the target-heading angle (
τ θ ) with the time to passage of the target (
τ p) when approaching the target. Accordingly, both
τ θ and
τ p should decrease smoothly with time as shown in
Figure 8a.
Figure 8b plots the time series of
τ θ and
τ p performance data, again computed from the recorded time series of the participant's position averaged across 12 participants. Positive
τ indicates that participants steered to close the relevant gap and negative
τ indicates that participants steered to enlarge the gap.
τ p starts at about 12 s and decreases to close to zero at the end of the trial, indicating that participants steered to shorten their distance to the target throughout the trial. In contrast,
τ θ starts at a negative value, consistent with the initial increase in the heading error (i.e., the initial enlarging of the target-heading angle,
Figure 5b), then increases and decreases abruptly, finally converging close to zero.
τ θ nevertheless spikes again after 3 s, which continues until the end of the trial. For both display conditions, the profile of
τ p shows a continual decrease and is in sharp contrast with the unstable profile of
τ θ . This indicates that participants did not try to equalize
τ θ with
τ p to travel on a curved path with changing curvature to the target (Fajen,
2001). On the contrary, the profiles of
τ θ and
τ p are consistent with the use of the heading strategy to steer toward the target via a straight path. Specifically, given
τ θ =
(Lee,
1974,
1976), where
θ is the target-heading angle (i.e., heading error), the spikes of
τ θ after 3 s are due to small changes in heading error (i.e., small
) when participants completed heading adjustment and were traveling on a straight path to the target.