Direct Calculation of Ellipse Overlap Areas for Force-Based Models of Pedestrian Dynamics Gary B. Hughes Mohcine Chraibi California Polytechnic Forschungszentrum Jülich State University, Institute for Advanced Simulation San Luis Obispo, CA USA Jülich, Germany
Pedestrian Spatial Aspect Radially Asymmetric • Velocity-Dependent • 01
Dynamic, Elliptical ‘Sensory Zone’ i j Semi-Axis Direction of Movement: Semi-Axis Lateral Swaying: Chraibi, M., Seyfried, A. and Schadschneider, A. (2010), “Generalized centrifugal- force model for pedestrian dynamics,” Physical Review E , 82 :4, p. 046111. 02
Driving Force i j Driving Force: Chraibi, M., Seyfried, A. and Schadschneider, A. (2010), “Generalized centrifugal- force model for pedestrian dynamics,” Physical Review E , 82 :4, p. 046111. 03
Repulsive Force i r i j d ij r j Repulsive Force: Chraibi, M., Seyfried, A. and Schadschneider, A. (2010), “Generalized centrifugal- force model for pedestrian dynamics,” Physical Review E , 82 :4, p. 046111. 04
Overlap and Oscillations i r i j d ij r j i r i j d ij r j 05
Generalized Centrifugal-Force Model i r i j d ij r j Overlapping Proportion: Chraibi, M., Seyfried, A. and Schadschneider, A. (2010), “Generalized centrifugal- force model for pedestrian dynamics,” Physical Review E , 82 :4, p. 046111. 06
Relative Position Classification 0 1: Transversal 2: Transversal at 2 3: Separated at 4 Points Points 0 1 4, 5: One Ellipse 6: Transversal at 2 Points 7: Externally Contained in the Other and Tangent at 1 Point Tangent 1 1 8: Internally 9: Internally 10, 11: Osculating and Tangent at 1 Point Tangent at 2 Points Hyperosculating 07
Overlap Area: Inscribed Polygons j i 08
Ellipse Area by Gauss-Green Formula � � cos cos sin sin cos sin 09
Ellipse Sector Area ( x 1 , y 1 ) θ 2 θ 1 φ ( x 2 , y 2 ) � Sector Area � 10
Ellipse Segment Area ( x 1 , y 1 ) θ 2 θ 1 φ ( x 2 , y 2 ) Segment Area 11
Ellipse Overlap Area Relative Position 1 Relative Position 2 Relative Position 6 Relative Positions 10, 11 Hughes, G.B., and Chraibi, M. (2014), “Calculating Ellipse Overlap Areas,” Computing and Visualization in Science 15 , pp. 291-301. 12
Intersection Points General Ellipse (Parametric) φ ( h , k ) ( 0 , 0 ) Hughes, G.B., and Chraibi, M. (2014), “Calculating Ellipse Overlap Areas,” Computing and Visualization in Science 15 , pp. 291-301. 13
Intersection Points General Ellipse (Implicit Polynomial) 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 Hughes, G.B., and Chraibi, M. (2014), “Calculating Ellipse Overlap Areas,” Computing and Visualization in Science 15 , pp. 291-301. 14
Intersection Points 2 2 1 1 1 1 0 1 1 1 2 2 2 1 2 2 0 2 2 2 Bézout determinant: Hughes, G.B., and Chraibi, M. (2014), “Calculating Ellipse Overlap Areas,” Computing and Visualization in Science 15 , pp. 291-301. 15
Run-Time Comparison Hughes, G.B., and Chraibi, M. (2014), “Calculating Ellipse Overlap Areas,” Computing and Visualization in Science 15 , pp. 291-301. 16
Validation: Spatial Randomness List of discrete points in a λ = point density within the continuous 2D domain: domain {( x 1 , y 1 ), ( x 2 , y 2 ), …, ( x n , y n )} = n / area Each point has a nearest neighbor at a specific Test Statistic with Standard distance: Normal Distribution: { D 1 , D 2 , …, D n } Random Sample of m ≈ 0.1 n Nearest-Neighbor Distances 17
Clark-Evans Test Statistic n = 100 discrete points Z = -4.16134 within [0, 1] Х [0, 1] Z = -0.205861 Deviations from a “2D Poisson Process”: More Regular (Z > 0) Z = +6.04962 More Clustered (Z < 0) 18
Spatial Randomness of Pedestrian Flow • Corridor, bidirectional flow • http://ped.fz-juelich.de/experiments/2013.06.19_Duesseldorf_Messe_BaSiGo/result/corrected/BI_CORR.zip 19 • bi_corr_400_b_02.txt
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