Formulation of time to collision for elliptical agents

Time to collision (TTC) is a key indicator of human locomotion, encompassing both pedestrian and vehicular traffic. Applications of the TTC concept span a wide spectrum from safety to traffic flow dynamics. However, there exists no generic formulation for TTC and its calculation depends highly on the context. Therefore, we would like to propose here a unified methodology that is generally applicable, regardless of the scope. This generalization is possible, if we introduce the assumption that the geometry of interacting agents can be approximated by arbitrary ellipses. Isotropic human crowds can be modelled as a special case when the ellipse is reduced to a circle, whereas the strongly anisotropic vehicular traffic is covered by the general case. The calculation of time to collision, requires knowledge of the distance of closest approach, a quantity that is not trivial to obtain with elliptical proxies. In short, the distance of closest approach is generally larger than the sum of the radii. For an analytical solution, we utilize a recent result from the physics of liquid crystals and then we proceed with the calculation of TTC for the massive pNEUMA dataset of naturalistic vehicular trajectories. Our findings show pronounced difference between cars and motorcycles, which is not typical in lane-free systems and possibly suggests a hybrid system.