A reference line is taken and studied over a
fixed period of time \(\Delta {t}\).

Using this method we can obtain the pedestrian flow \(J\) and the
velocity \(v_i\) of each pedestrian passing the reference line directly.
Thus, the flow over time \(\langle J \rangle_{\Delta t}\) and the time mean
velocity \(\langle v \rangle_{\Delta t}\)
can be calculated as

where \(N^{\Delta t}\) is the number of persons passing the reference line during the time interval \(\Delta {t}\).

\(t_N^{\Delta {t}}\) and \(t_1^{\Delta {t}}\) are the times when the first and last pedestrians pass the location in \(\Delta {t}\).

**Note:**Note: this time period can be different from \(\Delta {t}\).

The time mean velocity \(\langle v \rangle_{\Delta t}\) is defined as the mean value of the instantaneous velocities \(N^{\Delta t}\) pedestrians.

\(v_i(t)\) is calculated by use of the displacement of pedestrian \(i\) in a small time interval \(\Delta t^\prime\) around \(t\):

\[v_i(t)=\frac{\vec{x_i}(t+\Delta t^\prime/2)-\vec{x_i}(t-\Delta t^\prime/2))}{\Delta t^\prime}.\]