Fig 1.
Photograph of our experimental set-up to study crossing flows.
Agents participating in our experiment are shown in this photograph for a typical trial with crossing angle 120°. The three stages of the trial are shown here, viz. (a) before crossing (b) during crossing and (c) after crossing.
Fig 2.
Illustration of a trial of crossing flow from our experiments.
Traces of all the pedestrians involved for a typical trial has been shown with expected value of crossing angle equal to 60°. Three different instances of the trial has been shown here viz. before crossing (T1), during crossing (T2) and after crossing (T3). The actual values of time frames are T1 = 2.3 sec, T2 = 6.55 sec and T3 = 10.8 sec from the beginning of the trial. The two groups of pedestrians are denoted by blue and red dots. The tails behind each of the dots are basically the distances travelled by the pedestrians in previous 1.25 sec.
Fig 3.
Schematic representation for the formation of stripes and definition of orientation γ and physical separation λ of stripes.
Formation of stripes as a consequence of two groups crossing each other. In this schematic diagram the crossing angle between the two groups is α. The figure has been shown for three instances viz. before crossing (T1), during crossing (T2) and after crossing (T3). The two groups before crossing are denoted by blue and red squares, whose direction of motion is denoted by arrows of the same color. The green dotted arrow denotes the bisector of the crossing angle. The orientation γ of the stripes is measured counter-clockwise from the bisector. λ is the spatial separation between two stripes from the same group. For specific definitions of γ and λ see Fig 6.
Fig 4.
Pictorial representation of the edge-cutting algorithm.
Figure demonstrates the working process of the edge-cutting algorithm as a sequence of time. Here we show the process for two typical trials with α = 89.8° and α = 116.9°. Red and blue arrows indicate the direction of motion of the two groups represented by red and blue dots respectively. The lines connecting the dots in each of the groups are considered as the virtual bonds or ‘edges’ which are suppressed when cut by a pedestrian on the other group (see Materials and methods). The figures are shown for three instances, viz. Ti, (Ti + Tf)/2 and Tf. Ti and Tf denote the instances of time when the first and last edge-cut take place respectively. The edge-cutting process for the entire course of time for these two trials are shown as videos in supplementary materials (S2 and S3 Videos).
Fig 5.
Output of the pattern matching technique.
The panels on the left show fitting of pedestrian positions for the same trials shown in Fig 4. Fitting was done using the parametric sine curve f and the transformed coordinates (x′, y′). The blue and red parts of the plot represent the crests and troughs of the sine function respectively. The outputs of the fitting are ,
and
for the typical trial with α = 89.8° and
,
and
for the typical trial with α = 116.9°. (see Materials and methods) The panels on the right show variation of
as a function of
and
keeping ψ fixed to the value obtained from the fitting shown in the left panel. The region of occurrence of high values of
is shown in yellow. The function
was maximised to fit the sine function f. The maximum value of
for the trial with α = 89.8, as obtained by our optimisation procedure is 1.132, which occurred for
and
. Whereas, for the trial with α = 116.9° we obtained the maximum value of
, which occurred for
and
.
Fig 6.
Summary of different methods to estimate orientation γ and physical separation λ of the stripes.
The arrows in blue and red represents the direction of motion of the two groups. The schematic diagrams are shown for an arbitrary crossing angle. The dashed green arrow indicates the bisector of the crossing angle between the two group direction vectors. The lines in blue and red show the stripes from the two groups. γ is the angle between the direction of stripes and the bisector of crossing angle, always measured counterclockwise. (a) Estimation of orientation of the stripes and physical separation
between two stripes from the same group using the parametric sinusoidal fitting. In doing this calculation it was assumed that stripes from the two groups are parallel to each other and are equispaced, as shown in the figure. (b) Orientation of the stripes from the two groups when we assume that stripes from the same group are parallel to each other and are equispaced.
and
denote the orientation of stripes whose group direction vectors are left and right to the direction of bisector respectively. Using the same convention,
and
are the spatial separation between the stripes in those cases. This calculation was also done by fitting the two dimensional sine curve. (c) Estimation of orientation of the individual stripes that were found using the edge-cutting algorithm, for the two groups. γL or γR denote the orientation of individual stripes whose group direction vector is left or right to the direction of bisector respectively.
Fig 7.
Orientation γ of the stripes using different methods.
(a) Figure shows boxplots for obtained values of ,
,
from the pattern-matching technique and γL, γR from the edge-cutting algorithm. Outliers are shown by black dots. The dashed line corresponds to 90°. The boxplots were made using the various values of γ evaluated at the same time instant. This instant was chosen to be the time when the periodicity of the stripes from the two groups was best maintained (see Discussion). For detailed definitions of
,
,
, γL and γR see Fig 6 and Materials and methods. (b) Boxplots for the difference in obtained values of orientations
as estimated by separate-group analysis using pattern-matching technique. The values of
and
are the instantaneous values, which are shown in (a) (brown and orange bars).
Fig 8.
Time averaged orientation of stripes.
(a) Boxplots for and
.
and
are the orientations of stripes in a trial as obtained from separate-group analysis using pattern matching technique. (b) Boxplots for 〈γL〉t and 〈γR〉t, where γL and γR are estimated from the per-stripe analysis using edge-cutting algorithm.
Fig 9.
Boxplots for λ as estimated from the pattern matching technique.
λ is the spatial separation between two alternate stripes from the same group. (a) The boxplots were made over all the trials for obtained values of ,
and
at the instant when the periodicity of the two groups was best maintained.
,
and
are defined in Materials and methods. (b) Boxplots for the difference between obtained values of spatial separations from the whole-crowd and separate-group analyses under the pattern matching procedure. (c) Time-averaged difference of physical separation
, where
and
are the physical separations between stripes in a trial as estimated from the separate-group analysis using pattern matching technique.
Fig 10.
Width of stripes as a function of time.
Figure shows variation of stripe width as a function of time for all the stripes from a trial for two typical trials with (a) α = 89.8° and (b) α = 116.9°. Time t has been scaled as . Thus, t = Ti and t = Tf correspond to the scaled values of 0 and 1 respectively, which are shown by vertical dashed lines in the figure. For almost all the cases, we see that the width of the stripe attains a global minimum within the interval 0 and 1, which represents the ‘squeezing’ of stripes.
Fig 11.
Mean number of stripes emerging from a group.
Figure shows the variation of this quantity with crossing angle α. The mean was estimated over all the trials of our experiments. The number decreases with increasing α. The error-bars indicate the corresponding standard errors of mean.
Fig 12.
Examples of the edge-cutting process.
Edge-cutting process for two trials with (a) α = 63.8° and (b) α = 154.1°. The blue and red arrows denote the directions of motion for the two groups of pedestrians shown by blue and red dots respectively. The instances shown in this figure goes forward in time from (i) to (iii) and backward in time from (iii) to (v). In (i) the instance shown is Ti − 1, when all the edges within a group are intact. (ii) Shows the situation when the edges have started to cut and stripes are gradually being formed at (Ti + Tf)/2. (iii) Shows the situation at Tf + 1 when all probable edge-cuts have taken place and the stripes have completely been formed. (iv) and (v) shows the instances as in (ii) and (i) respectively but with the visualisation of all the stripes that are completely formed only after Tf.
Fig 13.
Boxplots for the maximising function of pattern matching technique.
(a) Boxplots for and
, where
is the maximising function of pattern matching procedure of whole-crowd analysis and
is the same with separate group-analysis.
and
are the maximum possible values of the maximising functions in these two cases, which are 2 and 1 respectively. (b) Boxplots for the time-averaged values of
and
.
Fig 14.
Accuracy of the pattern matching procedure.
Figure shows Normalised distributions for the distances of pedestrian positions from the crest or troughs of the fitted 2D sinusoid i.e. the residual errors. The distributions show a Gaussian peak at the origin for each α. The data were fitted according to a Gaussian curve and the fitted curves are shown by solid lines.
Fig 15.
Stripes for a typical trial of counter flow.
The red and blue dots denote the group of pedestrians that move along the direction of arrows shown in the same colors. The rectangular boxes in blue and red are minimum bounding boxes of the stripes that is used for per-stripe analysis. Individual stripe orientations γL and γR are also shown. The dashed green line indicates the bisector of the crossing angle. The dashed black lines enclosing each stripe are the convex hulls of pedestrian locations within that stripe. The stripe with 2 pedestrians was excluded from per-stripe analysis.
Table 1.
Summary of the experimental details.
Fig 16.
Schematic representation of the experimental set-up.
Figure shows the experimental set-up that we have constructed to study crossing flows of two groups of people. The dashed squares acted as visual references as the direction of motion for the participants. At the beginning of each trial, the participants were located in one these squares (S7 Fig) and were asked to reach the other side of the hall, crossing the other group.
Fig 17.
Schematic representation of the edge-cutting algorithm.
The ‘edge’ between the pedestrians P and Q belonging to the same group is cut by the pedestrian R belonging to the other group.
Fig 18.
Transformation of coordinates.
This diagram schematically represents the transformation given to the experimentally obtained data such that the transformed x-axis, i.e., x′ is along the bisector of the two group direction vectors. The arrows in blue and red indicate the two group direction vectors and the dotted green arrow indicates the bisector. θB is the angle between the bisector and the original x-axis. A clockwise rotation by an angle θB in this case makes the bisector as the new x-axis. The transformed axes x′ and y′ are shown by green arrows. The bold line in purple represents a stripe, which makes an angle γ (measured anti-clockwise) with the bisector of group direction or the x′-axis. The purpose of the pattern matching technique is to find out the orientation of stripes γ.
Fig 19.
Constructing the minimum bounding box of a stripe.
Two typical stripes are shown with aspect ratios (a) 0.999 and (b) 0.266. The red boxes denote the minimum bounding boxes and the polygons shown by black lines are the convex hulls of the points in the stripes. The stripe with the aspect ratio closer to 1 (a) is not suitable for the estimation of orientation. The stripe with the lower aspect ratio (b) is inclined at an angle of 76.01° with respect to the x-axis.