Evacuation around the obstacle
The route now uses two tangent segments and a circular arc, recalculated from each person’s current center after displacement. The full placement map has been rerun with this planner, including clearance from the walls and door edges.
At the selected point, 32/32 fresh layouts completed; 32/32 were faster than their matching control. The fixed-person times observed were 53.04–57.45 s. Very close placements can close the exit passages entirely; these outcomes remain visible.
Symmetric placement heatmap
Green means faster than the matched control; red means slower. Yellow means no change. The blank radius-wide border keeps the fixed circle inside the room. Select a tested point to inspect its time and recorded exit curve.
Colors interpolate the screening means. They show the change in time capped at 120 s. Hatching marks triangles touching an unfinished sample; it does not locate a proven jam boundary. The red crosshatched region blocks the point exit or both passages around the fixed circle. Small color changes can reflect layout variability; inspect the point intervals.
| Fixed center | Mean time | Change | Finished |
|---|
Run a fresh matched simulation
Generate 50 nonoverlapping circles with a reproducible seed, then simulate the same initial crowd with and without the fixed person. These two runs are separate from the heatmap’s averages.
No fixed person
Fixed person
Click a moving circle to inspect its route. The dashed circle shows center clearance, radius 1/8. Other movers and contact forces can alter the actual motion.
The corrected route
The physical fixed circle has radius 1/16. A moving center must avoid a circle with radius 1/16 + 1/16 = 1/8.
When the direct segment is obstructed, the planner compares the clockwise and counterclockwise routes and chooses the shorter feasible one. It follows the entry tangent, the circular arc, and the tangent toward the door. A circle visibility graph handles door-edge detours and rejects routes through walls.
This reference route is recalculated every time step. Finite time integration approximates motion along it; collisions and avoidance of moving neighbors can change the actual trajectory. The two tangent points are chosen by total route length.
Centerline placement and completion
Screening uses 12 independent layouts at each position. Error bars are pointwise 95% paired bootstrap intervals. The height range shown is 0.15–0.5; lower placements remain in the map and table.
Fresh confirmation
Nine candidates were chosen using a separate four-layout pilot, before examining these 32 fresh seeds. Intervals adjust for all nine candidates. Reflections do not add independent observations.
Time versus people out
Bands show variation across initial layouts, rather than confidence intervals for the mean. The plateau at 50 reflects the slowest run.
Individual clearance times
Each joined pair uses identical initial coordinates. Observed minima and maxima describe these samples.
Time-step and route accuracy
At 1.25 ms, the same 32 layouts average 55.04 s with the fixed person and 59.82 s without. Halving the step changes the mean paired saving by 0.323 s. The checks assess numerical sensitivity; they do not prove that every crowd trajectory converges identically.
Model and evidence
Room [0,2]², south opening x = 0.875–1.125, target (1,0). All physical radii are 1/16; preferred free speed is 1/8 unit/s. Fifty circles move; the fixed circle is never counted as an exit. A center reaching the target within 0.000001 unit is removed using a within-step event calculation.
- Frictionless, nonbouncy normal contacts. Equal-and-opposite pair impulses conserve pair momentum. Goal steering, walls and the immovable circle exchange momentum with the crowd, so total crowd momentum is not conserved.
- Static routes use exact circle tangencies and arcs. The time integration is finite-step. Moving-neighbor detours remain a local rule, refreshed every 0.15 s; obstructed driving relaxes over 0.25 s. Contact can change actual speed.
- Seeded sequential rejection sampling keeps initial circles inside the room and mutually clear. Only starts overlapping the proposed fixed circle are relocated. Every fixed-person run has a control using exactly the same adjusted initial layout.
- 219 screening positions, 12 independent seeds each, 5 ms. Nine confirmation candidates, 32 fresh seeds each, 2.5 ms. The selected point is repeated on the same 32 seeds at 1.25 ms. No results from the previous approximate planner are pooled into this map.
- Every reflected crowd is actually simulated. Positions and signed tie preferences reflect together. Screening, confirmation and smaller-step exit times and person IDs agree exactly with their reflection partners. Tiny coordinate-rounding changes altered some decimal-position pilot trajectories; those trials only informed candidate selection.
- Intervals use 50,000 paired bootstrap resamples. Confirmation uses a Bonferroni adjustment over nine candidates. Screening intervals are exploratory. The lowest observed mean is a finite-candidate result, not a proven continuous optimum.
- The map averages a Delaunay interpolation with its reflection before applying colors. All allowed centers lie inside the sampled hull. The 120 s cap keeps unfinished trials visible; their full clearance times are unknown.
- All eight formerly stalled layouts were rechecked: 8/8 completed with the revised navigation. This deliberately selected set diagnoses the revision; it does not estimate a general completion probability. Two new lateral stalls were rechecked to 240 s at both 5 and 2.5 ms; 0/4 diagnostic runs completed. These selected cases do not estimate a completion probability. This remains an idealized driven-circle model; results depend on its mobile avoidance and contact rules.
Map value = mean[min(T_fixed, 120 s) − min(T_control, 120 s)]