The optical angle (Tau) changes as a function of the distance to an object (Tau = invtan(ObjectSize / distance). It is argued that when regulating distance it is a common strategy to null the change in Tau (Tau dot).
I am trying to find evidence for this theory in cyclical backwards and forwards human-avatar locomotion. However, when analyzing Tau dot there is a problem, as due to the 'tangential' relationship between Tau and the distance to an object, the rate of change for a deviation of 1 meter is more when the object is close compared to far away. Moreover, a deviation of 1 meter closer to the object results in a bigger rate of change than a deviation of 1 meter away from the object. This results into a bias for using Tau dot to quantify success in distance keeping. How can I normalize tau dot so that it is not sensitive to the distance the object is perceived from?
(I've tried tau / tau dot, but that doesn't seem to work..)
Thanks for any suggestions!
Hi Rens,
One issue may be that tau (assuming you're using Dave Lee's variant of tau) is not the optical angle, but is rather ratio of the optical angle to its rate of change. (i.e. angle / angle-dot). Tau-dot is the rate of change of this ratio (dtau / dt). In terms of distance regulation, keeping tau-dot at a constant value can be a strategy to control the deceleration of a closing distance gap. This ratio is independent of the initial distance (greater initial distances can be reduced by greater initial velocity, resulting in a the same change of tau and constant tau-dot).
However, if the goal of your participants in the human-avatar co-ordination study is to to keep the perceived distance between them and the avatar constant, then tau-dot (and tau) may not be a sensible strategy, as the tau (time-to-contact) during constant distance is infinite/undefined (as rate-of-change is zero). If tau is infinite/undefined, then it makes no sense to talk of its rate-of-change (tau-dot). Perhaps tau may be used as an invariant to adjust deviations from a constant distance, but this does not imply that tau-dot will be kept constant, nor that distance to avatar should have no effect on this behaviour.
Hope that helps a bit,
Matthew
The ratio of rate of change of angle over angular size could perhaps do the job. Note that this ratio corresponds to the inverse of time-to-contact. Making this connection might lead you to choose another greek-alphabet letter to designate optical size: Since the seminal work of Lee (Perception, 1976), in the perception-action literature tau is generally associated with optical information about time-to-contact and its rate of change over time, tau-dot, with information about the sufficiency of current deceleration to avoid collision.
Hi Matthew and Reinoud,
Thanks for your replies. I see I will have to re-read some of Lee's work to better understand the initial intentions of his Tau. If I understand correctly 'Tau' is the angle / angle_dot. Tau_dot is then the rate of change of that ratio. Tau_dot is what has been shown to be kept constant when closing a gap.
And then time to contact is angle_dot / angle, which might be a good alternative, but to avoid confusion I shouldn't call it Tau.
This has been very helpful. Thanks!
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